| L(s) = 1 | + 4·3-s + 8·7-s + 6·9-s − 16-s + 12·17-s + 32·21-s − 8·25-s − 4·27-s + 24·29-s + 8·31-s − 16·37-s − 12·43-s − 8·47-s − 4·48-s + 32·49-s + 48·51-s − 8·53-s + 16·59-s + 8·61-s + 48·63-s + 8·67-s − 16·73-s − 32·75-s − 37·81-s + 16·83-s + 96·87-s − 56·89-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 2·9-s − 1/4·16-s + 2.91·17-s + 6.98·21-s − 8/5·25-s − 0.769·27-s + 4.45·29-s + 1.43·31-s − 2.63·37-s − 1.82·43-s − 1.16·47-s − 0.577·48-s + 32/7·49-s + 6.72·51-s − 1.09·53-s + 2.08·59-s + 1.02·61-s + 6.04·63-s + 0.977·67-s − 1.87·73-s − 3.69·75-s − 4.11·81-s + 1.75·83-s + 10.2·87-s − 5.93·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.824502088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.824502088\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7666 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 926 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 152 T^{3} - 62 T^{4} + 152 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 456 T^{3} + 6482 T^{4} + 456 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 472 T^{3} + 6898 T^{4} - 472 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 3458 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 80 T^{3} - 4574 T^{4} + 80 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1584 T^{3} + 19346 T^{4} - 1584 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 28 T + 366 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 40 T^{3} - 8414 T^{4} - 40 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.235919263959471435670853845723, −8.087846947698889139437704194872, −7.915688849310987436421230604929, −7.67869379192562561616960266002, −7.63164755779841768659563359090, −6.90911725299467861377485117544, −6.80001680515473308805609937200, −6.65413011823228063736139743228, −6.34672352747216141270714381340, −5.54116812885311822514635339121, −5.49296906175612882103997976984, −5.39056167601152267615243871118, −5.18758167396034018380986262370, −4.70665179338405949109924776657, −4.43692316008887322021167647127, −4.16038397790319089302222284634, −3.98631519311169405813218668025, −3.28145626014685230989178187682, −3.18615064066508699564036289858, −3.10079952322196621416810115887, −2.40607493791406122057582773128, −2.35247979678645319254139132156, −1.68392150978541173130160048334, −1.41808469667407532971803428356, −1.19324448058304920193204930480,
1.19324448058304920193204930480, 1.41808469667407532971803428356, 1.68392150978541173130160048334, 2.35247979678645319254139132156, 2.40607493791406122057582773128, 3.10079952322196621416810115887, 3.18615064066508699564036289858, 3.28145626014685230989178187682, 3.98631519311169405813218668025, 4.16038397790319089302222284634, 4.43692316008887322021167647127, 4.70665179338405949109924776657, 5.18758167396034018380986262370, 5.39056167601152267615243871118, 5.49296906175612882103997976984, 5.54116812885311822514635339121, 6.34672352747216141270714381340, 6.65413011823228063736139743228, 6.80001680515473308805609937200, 6.90911725299467861377485117544, 7.63164755779841768659563359090, 7.67869379192562561616960266002, 7.915688849310987436421230604929, 8.087846947698889139437704194872, 8.235919263959471435670853845723
