| L(s) = 1 | + 4·2-s + 12·4-s − 6·5-s + 7-s + 25·8-s − 24·10-s − 11-s − 8·13-s + 4·14-s + 45·16-s + 6·17-s − 19-s − 72·20-s − 4·22-s + 14·23-s + 25·25-s − 32·26-s + 12·28-s + 9·29-s + 3·31-s + 66·32-s + 24·34-s − 6·35-s + 12·37-s − 4·38-s − 150·40-s − 3·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 6·4-s − 2.68·5-s + 0.377·7-s + 8.83·8-s − 7.58·10-s − 0.301·11-s − 2.21·13-s + 1.06·14-s + 45/4·16-s + 1.45·17-s − 0.229·19-s − 16.0·20-s − 0.852·22-s + 2.91·23-s + 5·25-s − 6.27·26-s + 2.26·28-s + 1.67·29-s + 0.538·31-s + 11.6·32-s + 4.11·34-s − 1.01·35-s + 1.97·37-s − 0.648·38-s − 23.7·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.03228061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.03228061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p^{2} T + p^{2} T^{2} + 7 T^{3} - 21 T^{4} + 7 p T^{5} + p^{4} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 6 T + 11 T^{2} + 6 T^{3} + T^{4} + 6 p T^{5} + 11 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 8 T + 51 T^{2} + 244 T^{3} + 1049 T^{4} + 244 p T^{5} + 51 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 132 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 9 T + 7 T^{2} + 33 T^{3} + 400 T^{4} + 33 p T^{5} + 7 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 3 T - 27 T^{2} + 29 T^{3} + 900 T^{4} + 29 p T^{5} - 27 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 12 T + 57 T^{2} - 440 T^{3} + 3921 T^{4} - 440 p T^{5} + 57 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 3 T + 68 T^{2} + 81 T^{3} + 3055 T^{4} + 81 p T^{5} + 68 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 23 T + 202 T^{2} + 925 T^{3} + 4101 T^{4} + 925 p T^{5} + 202 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 160 p T^{5} + 43 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 3 T - 55 T^{2} - 57 T^{3} + 3364 T^{4} - 57 p T^{5} - 55 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 29 T^{2} - 240 T^{3} + 3001 T^{4} - 240 p T^{5} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 21 T + 95 T^{2} + 1371 T^{3} - 21536 T^{4} + 1371 p T^{5} + 95 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 4 T - 27 T^{2} + 500 T^{3} + 7301 T^{4} + 500 p T^{5} - 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 22 T + 225 T^{2} - 2402 T^{3} + 26159 T^{4} - 2402 p T^{5} + 225 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 17 T + 76 T^{2} + 1019 T^{3} - 19191 T^{4} + 1019 p T^{5} + 76 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 + 6 T - 61 T^{2} - 948 T^{3} + 229 T^{4} - 948 p T^{5} - 61 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.06061603740376869479139717241, −6.86861989254539340709474062377, −6.77712134924508694602715257828, −6.74486597400067788504987217534, −6.41401066780639407770903699851, −6.20511897678681777528378840130, −5.66283861682383227841124687221, −5.51141538238887306934474887787, −5.08496041897666157795569777859, −5.08469084509030989933315259519, −4.99075298688906689041703894827, −4.77250637324495139066941285854, −4.50616976434005615224285706875, −4.07187702317889266878213895630, −3.96935134380427099913842690830, −3.47179604253122515555837601275, −3.46520017265420928644145779666, −3.16586522056781641336620936753, −2.80804213221680473782275548595, −2.79036428106843959681131277779, −2.47237024591752638492640105621, −2.26781808758544159168998947588, −1.53286433284212050318867387138, −0.907438624422263675218261456862, −0.71248925022986609927474814306,
0.71248925022986609927474814306, 0.907438624422263675218261456862, 1.53286433284212050318867387138, 2.26781808758544159168998947588, 2.47237024591752638492640105621, 2.79036428106843959681131277779, 2.80804213221680473782275548595, 3.16586522056781641336620936753, 3.46520017265420928644145779666, 3.47179604253122515555837601275, 3.96935134380427099913842690830, 4.07187702317889266878213895630, 4.50616976434005615224285706875, 4.77250637324495139066941285854, 4.99075298688906689041703894827, 5.08469084509030989933315259519, 5.08496041897666157795569777859, 5.51141538238887306934474887787, 5.66283861682383227841124687221, 6.20511897678681777528378840130, 6.41401066780639407770903699851, 6.74486597400067788504987217534, 6.77712134924508694602715257828, 6.86861989254539340709474062377, 7.06061603740376869479139717241
