| L(s) = 1 | − 2.65·2-s − 1.47·3-s + 5.06·4-s − 0.123·5-s + 3.91·6-s − 2.04·7-s − 8.15·8-s − 0.835·9-s + 0.327·10-s − 1.27·11-s − 7.45·12-s + 1.33·13-s + 5.42·14-s + 0.181·15-s + 11.5·16-s + 2.50·17-s + 2.22·18-s − 0.794·19-s − 0.624·20-s + 3.00·21-s + 3.39·22-s + 6.36·23-s + 11.9·24-s − 4.98·25-s − 3.54·26-s + 5.64·27-s − 10.3·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.849·3-s + 2.53·4-s − 0.0551·5-s + 1.59·6-s − 0.771·7-s − 2.88·8-s − 0.278·9-s + 0.103·10-s − 0.384·11-s − 2.15·12-s + 0.369·13-s + 1.45·14-s + 0.0468·15-s + 2.88·16-s + 0.607·17-s + 0.523·18-s − 0.182·19-s − 0.139·20-s + 0.655·21-s + 0.723·22-s + 1.32·23-s + 2.44·24-s − 0.996·25-s − 0.694·26-s + 1.08·27-s − 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3775937772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3775937772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6037 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 + 0.123T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 + 0.794T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 9.30T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.55T + 83T^{2} \) |
| 89 | \( 1 + 4.05T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.968433806783362192457762606139, −7.66515055527370252782646078455, −6.68247781959569322235663491566, −6.20110305168716180562960822924, −5.70334163404499059900977091501, −4.55109258195489288208941732349, −3.11710866324353099272181565405, −2.65536770607419509998110840351, −1.29379248043765658236416359749, −0.49410442245623185326793183733,
0.49410442245623185326793183733, 1.29379248043765658236416359749, 2.65536770607419509998110840351, 3.11710866324353099272181565405, 4.55109258195489288208941732349, 5.70334163404499059900977091501, 6.20110305168716180562960822924, 6.68247781959569322235663491566, 7.66515055527370252782646078455, 7.968433806783362192457762606139
