| L(s) = 1 | − 21.3·2-s − 190.·3-s − 57.2·4-s − 625·5-s + 4.05e3·6-s + 1.21e4·8-s + 1.64e4·9-s + 1.33e4·10-s − 7.98e4·11-s + 1.08e4·12-s + 1.48e5·13-s + 1.18e5·15-s − 2.29e5·16-s + 1.96e5·17-s − 3.50e5·18-s + 7.43e5·19-s + 3.58e4·20-s + 1.70e6·22-s + 1.25e6·23-s − 2.30e6·24-s + 3.90e5·25-s − 3.16e6·26-s + 6.16e5·27-s + 4.19e6·29-s − 2.53e6·30-s + 6.78e6·31-s − 1.32e6·32-s + ⋯ |
| L(s) = 1 | − 0.942·2-s − 1.35·3-s − 0.111·4-s − 0.447·5-s + 1.27·6-s + 1.04·8-s + 0.835·9-s + 0.421·10-s − 1.64·11-s + 0.151·12-s + 1.43·13-s + 0.605·15-s − 0.875·16-s + 0.571·17-s − 0.787·18-s + 1.30·19-s + 0.0500·20-s + 1.55·22-s + 0.937·23-s − 1.41·24-s + 0.200·25-s − 1.35·26-s + 0.223·27-s + 1.10·29-s − 0.570·30-s + 1.32·31-s − 0.222·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6196865670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6196865670\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 21.3T + 512T^{2} \) |
| 3 | \( 1 + 190.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 7.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.48e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.25e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.11e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.48e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.28e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.27e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.96e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.94e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.48e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.47e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.78e8T + 7.60e17T^{2} \) |
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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
−10.58466336809035872776883225345, −9.723682495114998666550511379996, −8.395615182275628391378614358258, −7.77697270582991003808467637426, −6.57083480324910485486797730776, −5.38233996642255129462923956109, −4.69891766918313602830092734752, −3.11427068112900289440307543943, −1.13938210663431742967060272055, −0.56820656039121912680318805878,
0.56820656039121912680318805878, 1.13938210663431742967060272055, 3.11427068112900289440307543943, 4.69891766918313602830092734752, 5.38233996642255129462923956109, 6.57083480324910485486797730776, 7.77697270582991003808467637426, 8.395615182275628391378614358258, 9.723682495114998666550511379996, 10.58466336809035872776883225345
