| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 6·11-s − 12-s + 2·13-s + 15-s + 16-s − 4·17-s − 18-s + 4·19-s − 20-s − 6·22-s − 6·23-s + 24-s − 4·25-s − 2·26-s − 27-s + 9·29-s − 30-s + 5·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 1.27·22-s − 1.25·23-s + 0.204·24-s − 4/5·25-s − 0.392·26-s − 0.192·27-s + 1.67·29-s − 0.182·30-s + 0.898·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.02607808103437, −14.16019755823128, −13.82390299700132, −13.41358857859047, −12.33683705950256, −12.05992586156763, −11.72686662126655, −11.26145015869525, −10.73895202965786, −9.999877677346978, −9.656070517668108, −9.133965100665660, −8.379036336970816, −8.146144393018147, −7.411600799036259, −6.617038167193949, −6.411578156669428, −6.042414262599263, −4.909285208154251, −4.551001446051808, −3.676144345684372, −3.382671823735697, −2.232899502412355, −1.520184236093810, −0.9300833672677698, 0,
0.9300833672677698, 1.520184236093810, 2.232899502412355, 3.382671823735697, 3.676144345684372, 4.551001446051808, 4.909285208154251, 6.042414262599263, 6.411578156669428, 6.617038167193949, 7.411600799036259, 8.146144393018147, 8.379036336970816, 9.133965100665660, 9.656070517668108, 9.999877677346978, 10.73895202965786, 11.26145015869525, 11.72686662126655, 12.05992586156763, 12.33683705950256, 13.41358857859047, 13.82390299700132, 14.16019755823128, 15.02607808103437
