| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 11-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 20-s + 22-s + 25-s − 2·26-s − 4·28-s + 2·29-s + 8·31-s − 32-s − 2·34-s + 4·35-s + 10·37-s + 40-s − 10·41-s − 44-s + 9·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.676·35-s + 1.64·37-s + 0.158·40-s − 1.56·41-s − 0.150·44-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.78099998904700, −12.15436314165772, −11.91793238619662, −11.30746352134092, −10.98998317146445, −10.21547856582715, −9.955907333992355, −9.836963475559254, −9.076566892231846, −8.630074579945192, −8.264596499331034, −7.797743576675554, −7.218084056795522, −6.697764918900218, −6.483818132316794, −5.880682790938621, −5.438473474782995, −4.741586503868964, −4.067023969332511, −3.606840652505575, −3.119019769167093, −2.630411360737084, −2.120889799423661, −1.053011350923258, −0.7699287888029883, 0,
0.7699287888029883, 1.053011350923258, 2.120889799423661, 2.630411360737084, 3.119019769167093, 3.606840652505575, 4.067023969332511, 4.741586503868964, 5.438473474782995, 5.880682790938621, 6.483818132316794, 6.697764918900218, 7.218084056795522, 7.797743576675554, 8.264596499331034, 8.630074579945192, 9.076566892231846, 9.836963475559254, 9.955907333992355, 10.21547856582715, 10.98998317146445, 11.30746352134092, 11.91793238619662, 12.15436314165772, 12.78099998904700
