| L(s) = 1 | + 0.220·2-s − 1.95·4-s − 3.98·5-s + 0.113·7-s − 0.871·8-s − 0.878·10-s − 4.38·13-s + 0.0251·14-s + 3.71·16-s + 5.39·17-s − 4.99·19-s + 7.76·20-s − 5.08·23-s + 10.8·25-s − 0.967·26-s − 0.222·28-s + 1.72·29-s − 2.61·31-s + 2.56·32-s + 1.18·34-s − 0.453·35-s + 2.23·37-s − 1.10·38-s + 3.46·40-s − 9.28·41-s − 12.5·43-s − 1.12·46-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.975·4-s − 1.78·5-s + 0.0430·7-s − 0.308·8-s − 0.277·10-s − 1.21·13-s + 0.00671·14-s + 0.927·16-s + 1.30·17-s − 1.14·19-s + 1.73·20-s − 1.06·23-s + 2.16·25-s − 0.189·26-s − 0.0420·28-s + 0.319·29-s − 0.469·31-s + 0.452·32-s + 0.204·34-s − 0.0766·35-s + 0.367·37-s − 0.178·38-s + 0.548·40-s − 1.45·41-s − 1.92·43-s − 0.165·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1036650519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1036650519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.220T + 2T^{2} \) |
| 5 | \( 1 + 3.98T + 5T^{2} \) |
| 7 | \( 1 - 0.113T + 7T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 5.39T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 + 1.48T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.86756806823180797638411455385, −7.16385171190339187489419458842, −6.36713839546595193023969281996, −5.35884751334812490343548177528, −4.73496154064792023250658777760, −4.26489308796121353912411724958, −3.50468468348157229945472326495, −3.00332801802038508101939060454, −1.56038004682967233897071865579, −0.15149664865857680735774581048,
0.15149664865857680735774581048, 1.56038004682967233897071865579, 3.00332801802038508101939060454, 3.50468468348157229945472326495, 4.26489308796121353912411724958, 4.73496154064792023250658777760, 5.35884751334812490343548177528, 6.36713839546595193023969281996, 7.16385171190339187489419458842, 7.86756806823180797638411455385
