| L(s) = 1 | + 4·2-s − 4·3-s + 10·4-s + 4·5-s − 16·6-s + 20·8-s + 10·9-s + 16·10-s + 6·11-s − 40·12-s − 6·13-s − 16·15-s + 35·16-s + 10·17-s + 40·18-s + 4·19-s + 40·20-s + 24·22-s + 4·23-s − 80·24-s + 6·25-s − 24·26-s − 20·27-s + 6·29-s − 64·30-s − 2·31-s + 56·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s + 5.05·10-s + 1.80·11-s − 11.5·12-s − 1.66·13-s − 4.13·15-s + 35/4·16-s + 2.42·17-s + 9.42·18-s + 0.917·19-s + 8.94·20-s + 5.11·22-s + 0.834·23-s − 16.3·24-s + 6/5·25-s − 4.70·26-s − 3.84·27-s + 1.11·29-s − 11.6·30-s − 0.359·31-s + 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(78.48853385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(78.48853385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 25 T^{2} - 42 T^{3} + 120 T^{4} - 42 p T^{5} + 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 51 T^{2} + 198 T^{3} + 986 T^{4} + 198 p T^{5} + 51 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 83 T^{2} - 26 p T^{3} + 2098 T^{4} - 26 p^{2} T^{5} + 83 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 46 T^{2} - 196 T^{3} + 1090 T^{4} - 196 p T^{5} + 46 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 115 T^{2} - 498 T^{3} + 5004 T^{4} - 498 p T^{5} + 115 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 57 T^{2} + 406 T^{3} + 1424 T^{4} + 406 p T^{5} + 57 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 90 T^{2} + 72 T^{3} + 4058 T^{4} + 72 p T^{5} + 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 134 T^{2} + 460 T^{3} + 7690 T^{4} + 460 p T^{5} + 134 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 192 T^{2} - 28 p T^{3} + 7118 T^{4} - 28 p^{2} T^{5} + 192 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 113 T^{2} + 910 T^{3} + 7552 T^{4} + 910 p T^{5} + 113 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 154 T^{2} + 48 T^{3} + 11259 T^{4} + 48 p T^{5} + 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 187 T^{2} + 846 T^{3} + 15216 T^{4} + 846 p T^{5} + 187 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T^{2} - 384 T^{3} + 2870 T^{4} - 384 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 18 T + 169 T^{2} - 630 T^{3} + 2472 T^{4} - 630 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 42 T + 931 T^{2} - 13278 T^{3} + 132774 T^{4} - 13278 p T^{5} + 931 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 237 T^{2} + 1410 T^{3} + 23912 T^{4} + 1410 p T^{5} + 237 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 315 T^{2} + 1398 T^{3} + 37304 T^{4} + 1398 p T^{5} + 315 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 241 T^{2} - 2270 T^{3} + 27480 T^{4} - 2270 p T^{5} + 241 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 250 T^{2} - 504 T^{3} + 28026 T^{4} - 504 p T^{5} + 250 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 363 T^{2} - 2702 T^{3} + 51668 T^{4} - 2702 p T^{5} + 363 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.55258139307146897379593545920, −5.37265541491018826049114210095, −5.16585903080697192055642074753, −5.14870678305903803835405682126, −4.91570640292538884640912245187, −4.81564802158585044038272309813, −4.66747981985958497956036189310, −4.46570765178588112676024831118, −4.22125427862921388386670611142, −3.82063192266163779815895466871, −3.72133784711388178524495117433, −3.63608417891334765703529824066, −3.53100962178339838323815095959, −3.13435109206786038709475851728, −2.85648778612884278491622473793, −2.68727620987730109955211681675, −2.62223104914888469049735687188, −2.03259966197757748482384965238, −1.94041341659148928942914735835, −1.82779356968270071136194532440, −1.67451169162026248380000110977, −1.08288858398227914339660530080, −0.929069250681380349996872931449, −0.843739528948761683587663889322, −0.61548315087750233909378414294,
0.61548315087750233909378414294, 0.843739528948761683587663889322, 0.929069250681380349996872931449, 1.08288858398227914339660530080, 1.67451169162026248380000110977, 1.82779356968270071136194532440, 1.94041341659148928942914735835, 2.03259966197757748482384965238, 2.62223104914888469049735687188, 2.68727620987730109955211681675, 2.85648778612884278491622473793, 3.13435109206786038709475851728, 3.53100962178339838323815095959, 3.63608417891334765703529824066, 3.72133784711388178524495117433, 3.82063192266163779815895466871, 4.22125427862921388386670611142, 4.46570765178588112676024831118, 4.66747981985958497956036189310, 4.81564802158585044038272309813, 4.91570640292538884640912245187, 5.14870678305903803835405682126, 5.16585903080697192055642074753, 5.37265541491018826049114210095, 5.55258139307146897379593545920
