| L(s) = 1 | − 5·9-s − 12·11-s + 16·19-s − 12·29-s − 4·31-s − 12·41-s − 11·49-s + 2·61-s − 8·79-s + 9·81-s − 6·89-s + 60·99-s + 6·101-s + 34·109-s + 58·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s − 80·171-s + 173-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 3.61·11-s + 3.67·19-s − 2.22·29-s − 0.718·31-s − 1.87·41-s − 1.57·49-s + 0.256·61-s − 0.900·79-s + 81-s − 0.635·89-s + 6.03·99-s + 0.597·101-s + 3.25·109-s + 5.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s − 6.11·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1577051060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1577051060\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.59029294036418735482971328542, −7.49935966474912430085120076445, −7.06370523361702915577230407559, −6.98605285245511176583640423195, −6.61210937356520402815524906658, −6.32452547653195436148859367660, −5.86278475541282963043007932529, −5.65052428073291418042029720139, −5.51708881811588923150174182914, −5.41237621230457878230579942949, −5.30768511409901177601284251375, −4.98274831252056889652300491816, −4.85594022073743344940773153822, −4.43450097858714459320649525872, −4.02787335606104075141367167746, −3.44523461624446216444515653620, −3.33116525699630870877050130879, −3.22838470323681510562797052722, −3.00430670907121122626317071299, −2.57691839692257883958147439744, −2.43051744339291963771087778339, −1.82369861939299468177614442616, −1.67106804890505790399825074717, −0.838381786191964814041901911863, −0.12857275858498555271702710163,
0.12857275858498555271702710163, 0.838381786191964814041901911863, 1.67106804890505790399825074717, 1.82369861939299468177614442616, 2.43051744339291963771087778339, 2.57691839692257883958147439744, 3.00430670907121122626317071299, 3.22838470323681510562797052722, 3.33116525699630870877050130879, 3.44523461624446216444515653620, 4.02787335606104075141367167746, 4.43450097858714459320649525872, 4.85594022073743344940773153822, 4.98274831252056889652300491816, 5.30768511409901177601284251375, 5.41237621230457878230579942949, 5.51708881811588923150174182914, 5.65052428073291418042029720139, 5.86278475541282963043007932529, 6.32452547653195436148859367660, 6.61210937356520402815524906658, 6.98605285245511176583640423195, 7.06370523361702915577230407559, 7.49935966474912430085120076445, 7.59029294036418735482971328542
