| L(s) = 1 | + 2.74·2-s − 1.24·3-s + 5.54·4-s − 3.06·5-s − 3.41·6-s − 3.29·7-s + 9.75·8-s − 1.45·9-s − 8.43·10-s + 0.325·11-s − 6.89·12-s − 9.05·14-s + 3.81·15-s + 15.6·16-s + 4.63·17-s − 3.99·18-s − 7.31·19-s − 17.0·20-s + 4.09·21-s + 0.894·22-s + 6.13·23-s − 12.1·24-s + 4.41·25-s + 5.53·27-s − 18.2·28-s + 4.09·29-s + 10.4·30-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.717·3-s + 2.77·4-s − 1.37·5-s − 1.39·6-s − 1.24·7-s + 3.44·8-s − 0.484·9-s − 2.66·10-s + 0.0982·11-s − 1.99·12-s − 2.41·14-s + 0.985·15-s + 3.92·16-s + 1.12·17-s − 0.941·18-s − 1.67·19-s − 3.80·20-s + 0.893·21-s + 0.190·22-s + 1.28·23-s − 2.47·24-s + 0.883·25-s + 1.06·27-s − 3.45·28-s + 0.760·29-s + 1.91·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.455383941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.455383941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 0.325T + 11T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 6.50T + 41T^{2} \) |
| 43 | \( 1 - 0.943T + 43T^{2} \) |
| 47 | \( 1 + 7.06T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.690T + 59T^{2} \) |
| 61 | \( 1 - 3.13T + 61T^{2} \) |
| 67 | \( 1 - 9.72T + 67T^{2} \) |
| 71 | \( 1 + 3.47T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 - 8.52T + 97T^{2} \) |
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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
−7.892377450975534540654128588544, −7.00725494406103944919596009793, −6.53543068585336055572844947976, −6.01187703128456222561045011181, −5.16892474431084608540569627431, −4.53386376940567387181664716485, −3.73924688769699155236180520832, −3.23397471910173777812705310308, −2.49540242233894262805134765407, −0.75224337555261820306456249315,
0.75224337555261820306456249315, 2.49540242233894262805134765407, 3.23397471910173777812705310308, 3.73924688769699155236180520832, 4.53386376940567387181664716485, 5.16892474431084608540569627431, 6.01187703128456222561045011181, 6.53543068585336055572844947976, 7.00725494406103944919596009793, 7.892377450975534540654128588544
