L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 5·11-s + 12-s − 7·13-s − 14-s + 15-s + 16-s − 18-s − 19-s + 20-s + 21-s + 5·22-s − 3·23-s − 24-s + 25-s + 7·26-s + 27-s + 28-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s − 1.94·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 1.37·26-s + 0.192·27-s + 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.44835917855403, −12.84268923145652, −12.72825698506650, −12.00632394204639, −11.64796141500266, −10.76955328643907, −10.65013589042426, −10.02460253171417, −9.652288629772221, −9.326746758648319, −8.659615293011733, −8.101361867303186, −7.710168538469561, −7.472942028030411, −6.832952785539924, −6.243674864240308, −5.551652322276386, −5.027397051127129, −4.743990501393776, −3.892892004689704, −3.155828418621859, −2.509670085234682, −2.251262916071371, −1.754996273621146, −0.7047278583819732, 0,
0.7047278583819732, 1.754996273621146, 2.251262916071371, 2.509670085234682, 3.155828418621859, 3.892892004689704, 4.743990501393776, 5.027397051127129, 5.551652322276386, 6.243674864240308, 6.832952785539924, 7.472942028030411, 7.710168538469561, 8.101361867303186, 8.659615293011733, 9.326746758648319, 9.652288629772221, 10.02460253171417, 10.65013589042426, 10.76955328643907, 11.64796141500266, 12.00632394204639, 12.72825698506650, 12.84268923145652, 13.44835917855403