| L(s) = 1 | − 1.58·2-s − 3-s + 0.516·4-s + 5-s + 1.58·6-s + 5.23·7-s + 2.35·8-s + 9-s − 1.58·10-s − 3.22·11-s − 0.516·12-s + 13-s − 8.30·14-s − 15-s − 4.76·16-s + 1.09·17-s − 1.58·18-s + 6.84·19-s + 0.516·20-s − 5.23·21-s + 5.12·22-s − 7.52·23-s − 2.35·24-s + 25-s − 1.58·26-s − 27-s + 2.70·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.577·3-s + 0.258·4-s + 0.447·5-s + 0.647·6-s + 1.97·7-s + 0.831·8-s + 0.333·9-s − 0.501·10-s − 0.973·11-s − 0.149·12-s + 0.277·13-s − 2.21·14-s − 0.258·15-s − 1.19·16-s + 0.265·17-s − 0.373·18-s + 1.57·19-s + 0.115·20-s − 1.14·21-s + 1.09·22-s − 1.56·23-s − 0.480·24-s + 0.200·25-s − 0.311·26-s − 0.192·27-s + 0.510·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165007516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165007516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 17 | \( 1 - 1.09T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 37 | \( 1 + 0.0292T + 37T^{2} \) |
| 41 | \( 1 + 8.84T + 41T^{2} \) |
| 43 | \( 1 + 8.11T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 0.301T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.141506551119803896245053263612, −7.66795939023171600126528183066, −6.94029717699007734461681753014, −5.84019127383943787778182062457, −5.01194570591245517743637286387, −4.91323603188046293943678474684, −3.71861454834087912428788572770, −2.24405309145399855263627865438, −1.58275541735177675536128248705, −0.75131317984977275581917636377,
0.75131317984977275581917636377, 1.58275541735177675536128248705, 2.24405309145399855263627865438, 3.71861454834087912428788572770, 4.91323603188046293943678474684, 5.01194570591245517743637286387, 5.84019127383943787778182062457, 6.94029717699007734461681753014, 7.66795939023171600126528183066, 8.141506551119803896245053263612
