| L(s) = 1 | + 0.895·2-s − 0.415·3-s − 1.19·4-s − 5-s − 0.372·6-s + 2.78·7-s − 2.86·8-s − 2.82·9-s − 0.895·10-s − 5.25·11-s + 0.498·12-s + 1.97·13-s + 2.49·14-s + 0.415·15-s − 0.168·16-s + 6.98·17-s − 2.53·18-s + 8.09·19-s + 1.19·20-s − 1.15·21-s − 4.70·22-s + 3.31·23-s + 1.19·24-s + 25-s + 1.76·26-s + 2.42·27-s − 3.33·28-s + ⋯ |
| L(s) = 1 | + 0.633·2-s − 0.240·3-s − 0.599·4-s − 0.447·5-s − 0.152·6-s + 1.05·7-s − 1.01·8-s − 0.942·9-s − 0.283·10-s − 1.58·11-s + 0.143·12-s + 0.546·13-s + 0.666·14-s + 0.107·15-s − 0.0420·16-s + 1.69·17-s − 0.596·18-s + 1.85·19-s + 0.267·20-s − 0.252·21-s − 1.00·22-s + 0.690·23-s + 0.243·24-s + 0.200·25-s + 0.346·26-s + 0.466·27-s − 0.630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536788065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536788065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.895T + 2T^{2} \) |
| 3 | \( 1 + 0.415T + 3T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 + 6.04T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 - 8.17T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−9.747202885137889890185473561770, −8.813107275702906583725912807164, −7.918251264803500655584745539911, −7.63782393768294245040900912173, −5.94742369656559120249727683653, −5.24483124634825567234352822279, −4.94904993215484609554049198349, −3.53311663753622917214829820666, −2.86194338651279048927465013201, −0.873540940185277372826835377312,
0.873540940185277372826835377312, 2.86194338651279048927465013201, 3.53311663753622917214829820666, 4.94904993215484609554049198349, 5.24483124634825567234352822279, 5.94742369656559120249727683653, 7.63782393768294245040900912173, 7.918251264803500655584745539911, 8.813107275702906583725912807164, 9.747202885137889890185473561770
