| L(s) = 1 | + 2·2-s + 4·4-s + 12·5-s + 7·7-s + 8·8-s + 24·10-s + 60·11-s + 74·13-s + 14·14-s + 16·16-s + 36·17-s + 19·19-s + 48·20-s + 120·22-s + 192·23-s + 19·25-s + 148·26-s + 28·28-s − 12·29-s − 160·31-s + 32·32-s + 72·34-s + 84·35-s + 254·37-s + 38·38-s + 96·40-s − 114·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.07·5-s + 0.377·7-s + 0.353·8-s + 0.758·10-s + 1.64·11-s + 1.57·13-s + 0.267·14-s + 1/4·16-s + 0.513·17-s + 0.229·19-s + 0.536·20-s + 1.16·22-s + 1.74·23-s + 0.151·25-s + 1.11·26-s + 0.188·28-s − 0.0768·29-s − 0.926·31-s + 0.176·32-s + 0.363·34-s + 0.405·35-s + 1.12·37-s + 0.162·38-s + 0.379·40-s − 0.434·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.636256627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.636256627\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 12 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 114 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 330 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 + 990 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 + 318 T + p^{3} T^{2} \) |
| 89 | \( 1 + 90 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−8.928936832581746994935869911157, −7.71308991553445238934030670248, −6.78177773820045851827650425775, −6.15929497227474814193352860271, −5.61031587652874073353769285422, −4.65143875642506014863927677545, −3.74814436363781196068326824099, −2.98628056763045086884057541026, −1.56418691519052784722297466425, −1.26615455567850596187763760256,
1.26615455567850596187763760256, 1.56418691519052784722297466425, 2.98628056763045086884057541026, 3.74814436363781196068326824099, 4.65143875642506014863927677545, 5.61031587652874073353769285422, 6.15929497227474814193352860271, 6.78177773820045851827650425775, 7.71308991553445238934030670248, 8.928936832581746994935869911157
