| L(s) = 1 | − 3·3-s + 4·5-s + 6·9-s + 11-s + 13-s − 12·15-s − 2·17-s + 6·19-s + 2·23-s + 11·25-s − 9·27-s + 29-s + 4·31-s − 3·33-s − 2·37-s − 3·39-s + 2·41-s − 4·43-s + 24·45-s + 2·47-s + 6·51-s − 12·53-s + 4·55-s − 18·57-s + 9·59-s + 5·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 2·9-s + 0.301·11-s + 0.277·13-s − 3.09·15-s − 0.485·17-s + 1.37·19-s + 0.417·23-s + 11/5·25-s − 1.73·27-s + 0.185·29-s + 0.718·31-s − 0.522·33-s − 0.328·37-s − 0.480·39-s + 0.312·41-s − 0.609·43-s + 3.57·45-s + 0.291·47-s + 0.840·51-s − 1.64·53-s + 0.539·55-s − 2.38·57-s + 1.17·59-s + 0.640·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909533337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909533337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.43769308118746924680842482643, −6.71829735458500977366619034385, −6.29743680307897387850043745353, −5.77068578533118252383591981370, −5.08554552526517151567165784178, −4.77518329500932741216790976989, −3.53506892864079249450392011561, −2.41073426599220571540915572802, −1.46813120706819175089842901463, −0.818644360895583031625407140138,
0.818644360895583031625407140138, 1.46813120706819175089842901463, 2.41073426599220571540915572802, 3.53506892864079249450392011561, 4.77518329500932741216790976989, 5.08554552526517151567165784178, 5.77068578533118252383591981370, 6.29743680307897387850043745353, 6.71829735458500977366619034385, 7.43769308118746924680842482643
