L(s) = 1 | − 2-s + 2.82·3-s + 4-s − 2.82·6-s − 8-s + 5.00·9-s − 11-s + 2.82·12-s − 2.82·13-s + 16-s + 2.82·17-s − 5.00·18-s + 5.65·19-s + 22-s + 8·23-s − 2.82·24-s − 5·25-s + 2.82·26-s + 5.65·27-s + 2·29-s + 8.48·31-s − 32-s − 2.82·33-s − 2.82·34-s + 5.00·36-s + 2·37-s − 5.65·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s − 1.15·6-s − 0.353·8-s + 1.66·9-s − 0.301·11-s + 0.816·12-s − 0.784·13-s + 0.250·16-s + 0.685·17-s − 1.17·18-s + 1.29·19-s + 0.213·22-s + 1.66·23-s − 0.577·24-s − 25-s + 0.554·26-s + 1.08·27-s + 0.371·29-s + 1.52·31-s − 0.176·32-s − 0.492·33-s − 0.485·34-s + 0.833·36-s + 0.328·37-s − 0.917·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.124644924\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.124644924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−9.820477154514011232455265934263, −8.997466491481587343659289507775, −8.326665340597870716505581679439, −7.51717193146548534533156801639, −7.12777322153850334744972367874, −5.66182867008759703605509091147, −4.44963573958408879388959489379, −3.12631326230837995645522018306, −2.65856383754343291430511836413, −1.28019551589000328596134817038,
1.28019551589000328596134817038, 2.65856383754343291430511836413, 3.12631326230837995645522018306, 4.44963573958408879388959489379, 5.66182867008759703605509091147, 7.12777322153850334744972367874, 7.51717193146548534533156801639, 8.326665340597870716505581679439, 8.997466491481587343659289507775, 9.820477154514011232455265934263