| L(s) = 1 | + 2.41·2-s + 0.414·3-s + 3.82·4-s − 5-s + 0.999·6-s + 4.41·8-s − 2.82·9-s − 2.41·10-s − 0.828·11-s + 1.58·12-s + 4.82·13-s − 0.414·15-s + 2.99·16-s − 4.82·17-s − 6.82·18-s − 2.82·19-s − 3.82·20-s − 1.99·22-s + 0.414·23-s + 1.82·24-s + 25-s + 11.6·26-s − 2.41·27-s − 29-s − 0.999·30-s + 6·31-s − 1.58·32-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.239·3-s + 1.91·4-s − 0.447·5-s + 0.408·6-s + 1.56·8-s − 0.942·9-s − 0.763·10-s − 0.249·11-s + 0.457·12-s + 1.33·13-s − 0.106·15-s + 0.749·16-s − 1.17·17-s − 1.60·18-s − 0.648·19-s − 0.856·20-s − 0.426·22-s + 0.0863·23-s + 0.373·24-s + 0.200·25-s + 2.28·26-s − 0.464·27-s − 0.185·29-s − 0.182·30-s + 1.07·31-s − 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.885971657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.885971657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 0.414T + 3T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.414T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 - 9.58T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−12.29676671814416835404592247026, −11.25636786429156791956982373293, −10.88633594007959368871955684454, −8.998931123466527088577383332889, −8.037320149620824264731133241529, −6.61232254239300780776978889431, −5.87255390820298938816884145913, −4.61104680320551662806530745336, −3.64049755444429123970037675569, −2.49591203604440861640967031540,
2.49591203604440861640967031540, 3.64049755444429123970037675569, 4.61104680320551662806530745336, 5.87255390820298938816884145913, 6.61232254239300780776978889431, 8.037320149620824264731133241529, 8.998931123466527088577383332889, 10.88633594007959368871955684454, 11.25636786429156791956982373293, 12.29676671814416835404592247026
