| L(s) = 1 | + 2·3-s − 4-s + 4·5-s + 2·7-s + 2·9-s + 8·11-s − 2·12-s − 6·13-s + 8·15-s + 16-s + 8·17-s − 6·19-s − 4·20-s + 4·21-s + 2·23-s + 11·25-s + 6·27-s − 2·28-s − 8·31-s + 16·33-s + 8·35-s − 2·36-s − 4·37-s − 12·39-s + 16·41-s − 8·43-s − 8·44-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 1.78·5-s + 0.755·7-s + 2/3·9-s + 2.41·11-s − 0.577·12-s − 1.66·13-s + 2.06·15-s + 1/4·16-s + 1.94·17-s − 1.37·19-s − 0.894·20-s + 0.872·21-s + 0.417·23-s + 11/5·25-s + 1.15·27-s − 0.377·28-s − 1.43·31-s + 2.78·33-s + 1.35·35-s − 1/3·36-s − 0.657·37-s − 1.92·39-s + 2.49·41-s − 1.21·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.158473925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.158473925\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.13437877922897863304937266141, −9.646815499063054186834074989111, −9.309452998086215824252051440427, −9.165207656492607888824824895076, −8.943578040880880718892194943399, −8.332793426325294711094520894368, −7.72806585794032034237529460478, −7.52650072287790908987560977419, −6.89818071742350343781913485886, −6.33052212114051699569758935580, −6.14856283336793334163892137571, −5.40521041381552564579876583560, −5.06110825928366752043910502046, −4.40441253507506312046910418189, −4.18129345392005184432558934934, −3.12006495634983073863131257904, −3.09221932734240264363910095724, −2.06983888160099076064200945735, −1.69351090553464892297101927579, −1.17944656040793476409418748702,
1.17944656040793476409418748702, 1.69351090553464892297101927579, 2.06983888160099076064200945735, 3.09221932734240264363910095724, 3.12006495634983073863131257904, 4.18129345392005184432558934934, 4.40441253507506312046910418189, 5.06110825928366752043910502046, 5.40521041381552564579876583560, 6.14856283336793334163892137571, 6.33052212114051699569758935580, 6.89818071742350343781913485886, 7.52650072287790908987560977419, 7.72806585794032034237529460478, 8.332793426325294711094520894368, 8.943578040880880718892194943399, 9.165207656492607888824824895076, 9.309452998086215824252051440427, 9.646815499063054186834074989111, 10.13437877922897863304937266141
