| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s − 2·9-s − 10-s − 2·11-s − 12-s + 13-s + 3·14-s − 15-s + 16-s − 5·17-s + 2·18-s + 19-s + 20-s + 3·21-s + 2·22-s + 24-s + 25-s − 26-s + 5·27-s − 3·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.471·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s + 0.426·22-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s − 0.566·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.01409127432552, −14.76061889119233, −13.75467255758611, −13.53567592368760, −12.81771800841831, −12.52007994935005, −11.78247175434167, −11.19660952001221, −10.83153562958346, −10.37616235161142, −9.672507840931141, −9.227782396602464, −8.893398195008350, −8.061634155981513, −7.622723772664344, −6.781545416497750, −6.296228534981574, −6.063865654800140, −5.318912420054716, −4.683507150409667, −3.811274640799003, −2.924835019361505, −2.634437171062838, −1.729697406248864, −0.6859645011353400, 0,
0.6859645011353400, 1.729697406248864, 2.634437171062838, 2.924835019361505, 3.811274640799003, 4.683507150409667, 5.318912420054716, 6.063865654800140, 6.296228534981574, 6.781545416497750, 7.622723772664344, 8.061634155981513, 8.893398195008350, 9.227782396602464, 9.672507840931141, 10.37616235161142, 10.83153562958346, 11.19660952001221, 11.78247175434167, 12.52007994935005, 12.81771800841831, 13.53567592368760, 13.75467255758611, 14.76061889119233, 15.01409127432552
