| L(s) = 1 | + 2-s + 1.25·3-s + 4-s − 5-s + 1.25·6-s + 3.22·7-s + 8-s − 1.41·9-s − 10-s + 5.03·11-s + 1.25·12-s + 2.54·13-s + 3.22·14-s − 1.25·15-s + 16-s + 0.741·17-s − 1.41·18-s + 8.15·19-s − 20-s + 4.05·21-s + 5.03·22-s − 0.280·23-s + 1.25·24-s + 25-s + 2.54·26-s − 5.55·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.727·3-s + 0.5·4-s − 0.447·5-s + 0.514·6-s + 1.21·7-s + 0.353·8-s − 0.470·9-s − 0.316·10-s + 1.51·11-s + 0.363·12-s + 0.705·13-s + 0.860·14-s − 0.325·15-s + 0.250·16-s + 0.179·17-s − 0.333·18-s + 1.87·19-s − 0.223·20-s + 0.885·21-s + 1.07·22-s − 0.0585·23-s + 0.257·24-s + 0.200·25-s + 0.498·26-s − 1.06·27-s + 0.608·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.185415862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.185415862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 - 0.741T + 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 23 | \( 1 + 0.280T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 8.36T + 31T^{2} \) |
| 37 | \( 1 + 0.170T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 0.251T + 53T^{2} \) |
| 59 | \( 1 - 5.26T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 5.09T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−8.047019142528507151614627377269, −7.41588901999912646263345202905, −6.73976154576729024549640277553, −5.75795245802170794173206895032, −5.19137158271029478310284933405, −4.29649623100110992638609098920, −3.58762971102483680652391999971, −3.10819602063057933565651886419, −1.86165886592142469431788459105, −1.17326654723905351682546467155,
1.17326654723905351682546467155, 1.86165886592142469431788459105, 3.10819602063057933565651886419, 3.58762971102483680652391999971, 4.29649623100110992638609098920, 5.19137158271029478310284933405, 5.75795245802170794173206895032, 6.73976154576729024549640277553, 7.41588901999912646263345202905, 8.047019142528507151614627377269
