| L(s) = 1 | − 3·4-s − 5-s − 6·9-s − 12·11-s + 5·16-s − 4·19-s + 3·20-s + 25-s − 2·29-s + 4·31-s + 18·36-s + 4·41-s + 36·44-s + 6·45-s − 10·49-s + 12·55-s − 16·59-s − 12·61-s − 3·64-s − 24·71-s + 12·76-s − 20·79-s − 5·80-s + 27·81-s + 36·89-s + 4·95-s + 72·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.447·5-s − 2·9-s − 3.61·11-s + 5/4·16-s − 0.917·19-s + 0.670·20-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 3·36-s + 0.624·41-s + 5.42·44-s + 0.894·45-s − 1.42·49-s + 1.61·55-s − 2.08·59-s − 1.53·61-s − 3/8·64-s − 2.84·71-s + 1.37·76-s − 2.25·79-s − 0.559·80-s + 3·81-s + 3.81·89-s + 0.410·95-s + 7.23·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 + T \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.947550273198981438091464686696, −8.402368279712544431210177649490, −8.252858064103973279037456381237, −7.64947192507391358342580089136, −7.52061109178487627462618377984, −6.01956800466047829467852819351, −6.01134697239891133882099417885, −5.27104240153111024683298781878, −4.71404743035249738453779867980, −4.64971827708119930945851978376, −3.35895406827147515192265858691, −2.95203182599403093553876063934, −2.37337806558837572700935795284, 0, 0,
2.37337806558837572700935795284, 2.95203182599403093553876063934, 3.35895406827147515192265858691, 4.64971827708119930945851978376, 4.71404743035249738453779867980, 5.27104240153111024683298781878, 6.01134697239891133882099417885, 6.01956800466047829467852819351, 7.52061109178487627462618377984, 7.64947192507391358342580089136, 8.252858064103973279037456381237, 8.402368279712544431210177649490, 8.947550273198981438091464686696
