| L(s) = 1 | − 2·2-s + 4·4-s + 7·5-s − 7·7-s − 8·8-s − 14·10-s + 17·11-s + 12·13-s + 14·14-s + 16·16-s + 38·17-s − 43·19-s + 28·20-s − 34·22-s + 131·23-s − 76·25-s − 24·26-s − 28·28-s + 160·29-s + 45·31-s − 32·32-s − 76·34-s − 49·35-s − 331·37-s + 86·38-s − 56·40-s − 111·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.626·5-s − 0.377·7-s − 0.353·8-s − 0.442·10-s + 0.465·11-s + 0.256·13-s + 0.267·14-s + 1/4·16-s + 0.542·17-s − 0.519·19-s + 0.313·20-s − 0.329·22-s + 1.18·23-s − 0.607·25-s − 0.181·26-s − 0.188·28-s + 1.02·29-s + 0.260·31-s − 0.176·32-s − 0.383·34-s − 0.236·35-s − 1.47·37-s + 0.367·38-s − 0.221·40-s − 0.422·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.513833874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513833874\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 17 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 - 131 T + p^{3} T^{2} \) |
| 29 | \( 1 - 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 45 T + p^{3} T^{2} \) |
| 37 | \( 1 + 331 T + p^{3} T^{2} \) |
| 41 | \( 1 + 111 T + p^{3} T^{2} \) |
| 43 | \( 1 - 230 T + p^{3} T^{2} \) |
| 47 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 53 | \( 1 - 396 T + p^{3} T^{2} \) |
| 59 | \( 1 - 214 T + p^{3} T^{2} \) |
| 61 | \( 1 - 768 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 - 551 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 390 T + p^{3} T^{2} \) |
| 83 | \( 1 - 440 T + p^{3} T^{2} \) |
| 89 | \( 1 - 105 T + p^{3} T^{2} \) |
| 97 | \( 1 - 304 T + p^{3} T^{2} \) |
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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
−10.68385485119613503566300198818, −9.992775179213121128210113356838, −9.127906286351567694877459202440, −8.369647940335319701601227440471, −7.11330208477243774812399161253, −6.33922289017477164872867849883, −5.27520675460473402310422871993, −3.67289520796133534732867935997, −2.32190912154338086011387358723, −0.931426166782279890743529621887,
0.931426166782279890743529621887, 2.32190912154338086011387358723, 3.67289520796133534732867935997, 5.27520675460473402310422871993, 6.33922289017477164872867849883, 7.11330208477243774812399161253, 8.369647940335319701601227440471, 9.127906286351567694877459202440, 9.992775179213121128210113356838, 10.68385485119613503566300198818
