| L(s) = 1 | − 3-s − 3·7-s + 9-s − 3·11-s − 13-s + 7·17-s − 8·19-s + 3·21-s − 4·23-s − 27-s + 3·29-s + 11·31-s + 3·33-s + 39-s − 2·41-s − 8·43-s + 9·47-s + 2·49-s − 7·51-s − 9·53-s + 8·57-s − 9·59-s − 61-s − 3·63-s + 5·67-s + 4·69-s − 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 1.69·17-s − 1.83·19-s + 0.654·21-s − 0.834·23-s − 0.192·27-s + 0.557·29-s + 1.97·31-s + 0.522·33-s + 0.160·39-s − 0.312·41-s − 1.21·43-s + 1.31·47-s + 2/7·49-s − 0.980·51-s − 1.23·53-s + 1.05·57-s − 1.17·59-s − 0.128·61-s − 0.377·63-s + 0.610·67-s + 0.481·69-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.41222600454448, −14.06172848602887, −13.32288991477208, −12.98795950630710, −12.44554532866661, −12.08705691254320, −11.67782184973227, −10.76755452735050, −10.33684528745102, −10.09395473779467, −9.647665733550768, −8.871198329997765, −8.182710763217545, −7.880697672879719, −7.177347040684166, −6.442651100461814, −6.205799677051385, −5.690185619158049, −4.878676228340173, −4.510029723507693, −3.679040220756485, −3.078100145031118, −2.525888155228745, −1.689324489672568, −0.6977153842888792, 0,
0.6977153842888792, 1.689324489672568, 2.525888155228745, 3.078100145031118, 3.679040220756485, 4.510029723507693, 4.878676228340173, 5.690185619158049, 6.205799677051385, 6.442651100461814, 7.177347040684166, 7.880697672879719, 8.182710763217545, 8.871198329997765, 9.647665733550768, 10.09395473779467, 10.33684528745102, 10.76755452735050, 11.67782184973227, 12.08705691254320, 12.44554532866661, 12.98795950630710, 13.32288991477208, 14.06172848602887, 14.41222600454448
