| L(s) = 1 | − 1.11·2-s + 3.24·3-s − 0.766·4-s + 5-s − 3.60·6-s + 2.31·7-s + 3.07·8-s + 7.51·9-s − 1.11·10-s + 11-s − 2.48·12-s − 0.104·13-s − 2.57·14-s + 3.24·15-s − 1.87·16-s + 3.59·17-s − 8.34·18-s + 3.68·19-s − 0.766·20-s + 7.51·21-s − 1.11·22-s + 7.11·23-s + 9.96·24-s + 25-s + 0.115·26-s + 14.6·27-s − 1.77·28-s + ⋯ |
| L(s) = 1 | − 0.785·2-s + 1.87·3-s − 0.383·4-s + 0.447·5-s − 1.47·6-s + 0.876·7-s + 1.08·8-s + 2.50·9-s − 0.351·10-s + 0.301·11-s − 0.717·12-s − 0.0289·13-s − 0.687·14-s + 0.837·15-s − 0.469·16-s + 0.873·17-s − 1.96·18-s + 0.846·19-s − 0.171·20-s + 1.64·21-s − 0.236·22-s + 1.48·23-s + 2.03·24-s + 0.200·25-s + 0.0227·26-s + 2.81·27-s − 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220487283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220487283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 - 3.24T + 3T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 13 | \( 1 + 0.104T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 0.187T + 31T^{2} \) |
| 37 | \( 1 + 2.99T + 37T^{2} \) |
| 41 | \( 1 + 9.01T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 + 0.802T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 8.85T + 71T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 8.39T + 83T^{2} \) |
| 89 | \( 1 - 2.19T + 89T^{2} \) |
| 97 | \( 1 + 0.769T + 97T^{2} \) |
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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
−8.324494147460861072604713799596, −8.145675501839549550617828276857, −7.35732016875580111070555713367, −6.68906575182019331506150878814, −5.06719315100310452879690674160, −4.73328942933591730448546781933, −3.54014558261497007310681176019, −2.97087303139986548138646878883, −1.66725177566375676701605661785, −1.30109536093984588273527779170,
1.30109536093984588273527779170, 1.66725177566375676701605661785, 2.97087303139986548138646878883, 3.54014558261497007310681176019, 4.73328942933591730448546781933, 5.06719315100310452879690674160, 6.68906575182019331506150878814, 7.35732016875580111070555713367, 8.145675501839549550617828276857, 8.324494147460861072604713799596
