| L(s) = 1 | − 2-s + 4-s + 0.448·5-s − 1.52·7-s − 8-s − 0.448·10-s − 0.412·11-s + 3.36·13-s + 1.52·14-s + 16-s + 3.96·17-s − 5.13·19-s + 0.448·20-s + 0.412·22-s − 2.63·23-s − 4.79·25-s − 3.36·26-s − 1.52·28-s − 2.15·29-s − 5.88·31-s − 32-s − 3.96·34-s − 0.686·35-s + 10.4·37-s + 5.13·38-s − 0.448·40-s − 5.89·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.200·5-s − 0.578·7-s − 0.353·8-s − 0.141·10-s − 0.124·11-s + 0.933·13-s + 0.408·14-s + 0.250·16-s + 0.961·17-s − 1.17·19-s + 0.100·20-s + 0.0878·22-s − 0.549·23-s − 0.959·25-s − 0.659·26-s − 0.289·28-s − 0.399·29-s − 1.05·31-s − 0.176·32-s − 0.680·34-s − 0.116·35-s + 1.72·37-s + 0.832·38-s − 0.0709·40-s − 0.920·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172255923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172255923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 0.448T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 0.412T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 2.63T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 + 5.88T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 0.410T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.960865904007374876264147644915, −7.23763747353153598443852685067, −6.38040381725908857039024478784, −5.99228815736715506511393989859, −5.25943166075589544539360532830, −4.01157719949299304487884685671, −3.54935941665052122806013791530, −2.47901431769468590366752572893, −1.72052007214366446129820805432, −0.59233792884023619034504634310,
0.59233792884023619034504634310, 1.72052007214366446129820805432, 2.47901431769468590366752572893, 3.54935941665052122806013791530, 4.01157719949299304487884685671, 5.25943166075589544539360532830, 5.99228815736715506511393989859, 6.38040381725908857039024478784, 7.23763747353153598443852685067, 7.960865904007374876264147644915
