| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 13·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 26·10-s + 11·11-s + 12·12-s − 67·13-s − 14·14-s − 39·15-s + 16·16-s + 8·17-s + 18·18-s + 21·19-s − 52·20-s − 21·21-s + 22·22-s − 194·23-s + 24·24-s + 44·25-s − 134·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.16·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.822·10-s + 0.301·11-s + 0.288·12-s − 1.42·13-s − 0.267·14-s − 0.671·15-s + 1/4·16-s + 0.114·17-s + 0.235·18-s + 0.253·19-s − 0.581·20-s − 0.218·21-s + 0.213·22-s − 1.75·23-s + 0.204·24-s + 0.351·25-s − 1.01·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 + 13 T + p^{3} T^{2} \) |
| 13 | \( 1 + 67 T + p^{3} T^{2} \) |
| 17 | \( 1 - 8 T + p^{3} T^{2} \) |
| 19 | \( 1 - 21 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 29 | \( 1 + 221 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 347 T + p^{3} T^{2} \) |
| 41 | \( 1 - 292 T + p^{3} T^{2} \) |
| 43 | \( 1 + 458 T + p^{3} T^{2} \) |
| 47 | \( 1 - 221 T + p^{3} T^{2} \) |
| 53 | \( 1 + 642 T + p^{3} T^{2} \) |
| 59 | \( 1 - 273 T + p^{3} T^{2} \) |
| 61 | \( 1 + 530 T + p^{3} T^{2} \) |
| 67 | \( 1 - 561 T + p^{3} T^{2} \) |
| 71 | \( 1 - 604 T + p^{3} T^{2} \) |
| 73 | \( 1 - 703 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 144 T + p^{3} T^{2} \) |
| 89 | \( 1 - 750 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1370 T + p^{3} T^{2} \) |
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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
−10.16578035332618344607212200209, −9.362559875366799087360062075473, −8.042801347268744024174521890850, −7.52091427786785616104914849464, −6.54155801570978699712565286659, −5.18539860695192908408691812831, −4.09342241777314052241806719853, −3.38807624007783473853024161630, −2.09491799671963314555185556744, 0,
2.09491799671963314555185556744, 3.38807624007783473853024161630, 4.09342241777314052241806719853, 5.18539860695192908408691812831, 6.54155801570978699712565286659, 7.52091427786785616104914849464, 8.042801347268744024174521890850, 9.362559875366799087360062075473, 10.16578035332618344607212200209
