| L(s) = 1 | − 0.327·2-s + 3-s − 1.89·4-s − 5-s − 0.327·6-s − 1.80·7-s + 1.27·8-s + 9-s + 0.327·10-s − 1.07·11-s − 1.89·12-s − 3.80·13-s + 0.592·14-s − 15-s + 3.36·16-s + 4.59·17-s − 0.327·18-s − 5.23·19-s + 1.89·20-s − 1.80·21-s + 0.350·22-s + 6.59·23-s + 1.27·24-s + 25-s + 1.24·26-s + 27-s + 3.41·28-s + ⋯ |
| L(s) = 1 | − 0.231·2-s + 0.577·3-s − 0.946·4-s − 0.447·5-s − 0.133·6-s − 0.682·7-s + 0.451·8-s + 0.333·9-s + 0.103·10-s − 0.322·11-s − 0.546·12-s − 1.05·13-s + 0.158·14-s − 0.258·15-s + 0.841·16-s + 1.11·17-s − 0.0772·18-s − 1.19·19-s + 0.423·20-s − 0.394·21-s + 0.0747·22-s + 1.37·23-s + 0.260·24-s + 0.200·25-s + 0.244·26-s + 0.192·27-s + 0.646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.327T + 2T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 6.59T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 + 3.33T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 + 1.54T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 + 4.35T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.908768024946419169458888667861, −7.22135716579278126848285723627, −6.43748871885200808722541682238, −5.40766829407526704495457715620, −4.72523338216047675431989621571, −4.04792682946735332988917383925, −3.20779141591208947947073199312, −2.53220028561952607525483851744, −1.10778048579500239511869077337, 0,
1.10778048579500239511869077337, 2.53220028561952607525483851744, 3.20779141591208947947073199312, 4.04792682946735332988917383925, 4.72523338216047675431989621571, 5.40766829407526704495457715620, 6.43748871885200808722541682238, 7.22135716579278126848285723627, 7.908768024946419169458888667861
