| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 3·11-s − 12-s + 2·13-s − 15-s − 16-s + 3·17-s − 18-s − 2·19-s + 20-s + 3·22-s + 3·24-s + 25-s − 2·26-s + 27-s + 10·29-s + 30-s − 4·31-s − 5·32-s − 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.639·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.718·31-s − 0.883·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.78527482768749, −12.20442613734057, −11.89288931760856, −11.05315734303154, −10.81043923929071, −10.31066074839104, −9.942808499585528, −9.574213595198914, −8.830851657725121, −8.638372424238843, −8.222059606698593, −7.771164435984261, −7.512128061932646, −6.776320624921388, −6.436430786635768, −5.618157560691385, −5.154656906639168, −4.662279926774194, −4.251088287167991, −3.557641412018966, −3.202319909942949, −2.634443758372170, −1.807396724228371, −1.391712824834253, −0.6496086348090133, 0,
0.6496086348090133, 1.391712824834253, 1.807396724228371, 2.634443758372170, 3.202319909942949, 3.557641412018966, 4.251088287167991, 4.662279926774194, 5.154656906639168, 5.618157560691385, 6.436430786635768, 6.776320624921388, 7.512128061932646, 7.771164435984261, 8.222059606698593, 8.638372424238843, 8.830851657725121, 9.574213595198914, 9.942808499585528, 10.31066074839104, 10.81043923929071, 11.05315734303154, 11.89288931760856, 12.20442613734057, 12.78527482768749
