| L(s) = 1 | − 2-s + 4-s + 2.82·5-s − 8-s − 3·9-s − 2.82·10-s + 11-s − 5.65·13-s + 16-s − 2.82·17-s + 3·18-s − 8.48·19-s + 2.82·20-s − 22-s − 8·23-s + 3.00·25-s + 5.65·26-s − 6·29-s + 8.48·31-s − 32-s + 2.82·34-s − 3·36-s − 6·37-s + 8.48·38-s − 2.82·40-s + 8.48·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26·5-s − 0.353·8-s − 9-s − 0.894·10-s + 0.301·11-s − 1.56·13-s + 0.250·16-s − 0.685·17-s + 0.707·18-s − 1.94·19-s + 0.632·20-s − 0.213·22-s − 1.66·23-s + 0.600·25-s + 1.10·26-s − 1.11·29-s + 1.52·31-s − 0.176·32-s + 0.485·34-s − 0.5·36-s − 0.986·37-s + 1.37·38-s − 0.447·40-s + 1.32·41-s − 0.609·43-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−9.530730185409156455610738588161, −8.734877997248979398883447111660, −8.039633612488981166741343372411, −6.85101474999731051432473448282, −6.17053049575869876704436871049, −5.42267652915547103492609807415, −4.18712210120360699910405087214, −2.46348138242626626833794123960, −2.08526057766359163412999320943, 0,
2.08526057766359163412999320943, 2.46348138242626626833794123960, 4.18712210120360699910405087214, 5.42267652915547103492609807415, 6.17053049575869876704436871049, 6.85101474999731051432473448282, 8.039633612488981166741343372411, 8.734877997248979398883447111660, 9.530730185409156455610738588161
