| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 2·13-s + 16-s − 6·17-s + 5·19-s − 20-s + 22-s + 3·23-s − 4·25-s − 2·26-s + 2·29-s − 5·31-s − 32-s + 6·34-s + 3·37-s − 5·38-s + 40-s + 3·41-s − 2·43-s − 44-s − 3·46-s − 10·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 1.14·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.371·29-s − 0.898·31-s − 0.176·32-s + 1.02·34-s + 0.493·37-s − 0.811·38-s + 0.158·40-s + 0.468·41-s − 0.304·43-s − 0.150·44-s − 0.442·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.551423168398543236619112644628, −7.75240721792102384696135904334, −7.14137669295085940317588036997, −6.34002059224990781957978985695, −5.45883024829548815631361397772, −4.46934491133839587358262773639, −3.50899548863681658313549736296, −2.55069351015900942552710835973, −1.37982935479584858529094782557, 0,
1.37982935479584858529094782557, 2.55069351015900942552710835973, 3.50899548863681658313549736296, 4.46934491133839587358262773639, 5.45883024829548815631361397772, 6.34002059224990781957978985695, 7.14137669295085940317588036997, 7.75240721792102384696135904334, 8.551423168398543236619112644628
