| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s + 11-s − 2·12-s + 13-s − 2·15-s − 4·16-s + 17-s + 2·18-s − 2·19-s + 4·20-s + 2·22-s + 4·23-s − 25-s + 2·26-s − 27-s − 4·30-s − 5·31-s − 8·32-s − 33-s + 2·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s + 0.894·20-s + 0.426·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.730·30-s − 0.898·31-s − 1.41·32-s − 0.174·33-s + 0.342·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.12699948624630, −14.73542120007616, −14.14058158081063, −13.63002858340630, −13.21777936745904, −12.82601021112162, −12.27779056185385, −11.65423007130614, −11.30826566037269, −10.65440041247929, −10.05407508797822, −9.487629566096687, −8.917862127854376, −8.260456340311266, −7.363211394971424, −6.597457357018208, −6.469128201720409, −5.686316731524634, −5.278815840761903, −4.837976814419278, −4.000577455120850, −3.550241006846177, −2.735467994651662, −1.999179460461132, −1.278678228768352, 0,
1.278678228768352, 1.999179460461132, 2.735467994651662, 3.550241006846177, 4.000577455120850, 4.837976814419278, 5.278815840761903, 5.686316731524634, 6.469128201720409, 6.597457357018208, 7.363211394971424, 8.260456340311266, 8.917862127854376, 9.487629566096687, 10.05407508797822, 10.65440041247929, 11.30826566037269, 11.65423007130614, 12.27779056185385, 12.82601021112162, 13.21777936745904, 13.63002858340630, 14.14058158081063, 14.73542120007616, 15.12699948624630
