| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s − 88.0·7-s − 64·8-s + 81·9-s − 100·10-s + 673.·11-s + 144·12-s + 382.·13-s + 352.·14-s + 225·15-s + 256·16-s + 1.55e3·17-s − 324·18-s + 361·19-s + 400·20-s − 792.·21-s − 2.69e3·22-s − 893.·23-s − 576·24-s + 625·25-s − 1.52e3·26-s + 729·27-s − 1.40e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.679·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.67·11-s + 0.288·12-s + 0.626·13-s + 0.480·14-s + 0.258·15-s + 0.250·16-s + 1.30·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.392·21-s − 1.18·22-s − 0.352·23-s − 0.204·24-s + 0.200·25-s − 0.443·26-s + 0.192·27-s − 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.489457863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489457863\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 + 88.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 673.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 382.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 893.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 474.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 249.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.37e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.00e5T + 8.58e9T^{2} \) |
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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
−9.720310621953929151428995911183, −9.181065105725501362441464940654, −8.412429070745202052139722810021, −7.31144338047617505955717758667, −6.50453214492779801457285343888, −5.65554553337824510103351427267, −3.93840482561963732131821276935, −3.16693787605141757273641782859, −1.78310336414862252472290422783, −0.891955783496050536116547079658,
0.891955783496050536116547079658, 1.78310336414862252472290422783, 3.16693787605141757273641782859, 3.93840482561963732131821276935, 5.65554553337824510103351427267, 6.50453214492779801457285343888, 7.31144338047617505955717758667, 8.412429070745202052139722810021, 9.181065105725501362441464940654, 9.720310621953929151428995911183
