| L(s) = 1 | + 2.20·2-s + 3-s + 2.85·4-s − 5-s + 2.20·6-s − 3.44·7-s + 1.88·8-s + 9-s − 2.20·10-s + 0.997·11-s + 2.85·12-s + 13-s − 7.59·14-s − 15-s − 1.56·16-s − 4.89·17-s + 2.20·18-s − 1.09·19-s − 2.85·20-s − 3.44·21-s + 2.19·22-s + 6.11·23-s + 1.88·24-s + 25-s + 2.20·26-s + 27-s − 9.84·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.577·3-s + 1.42·4-s − 0.447·5-s + 0.899·6-s − 1.30·7-s + 0.665·8-s + 0.333·9-s − 0.696·10-s + 0.300·11-s + 0.823·12-s + 0.277·13-s − 2.03·14-s − 0.258·15-s − 0.390·16-s − 1.18·17-s + 0.519·18-s − 0.250·19-s − 0.638·20-s − 0.752·21-s + 0.468·22-s + 1.27·23-s + 0.384·24-s + 0.200·25-s + 0.432·26-s + 0.192·27-s − 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 0.997T + 11T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 - 6.11T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 + 9.17T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 + 7.15T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 0.442T + 71T^{2} \) |
| 73 | \( 1 - 4.55T + 73T^{2} \) |
| 79 | \( 1 - 0.866T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.31693985110419627784309830236, −6.83704582334305772487332733025, −6.30669914534200722044857501360, −5.49902374106986148619709497199, −4.59262049442010086213140854358, −4.01857204132143657056562431371, −3.27394640911317126436954755060, −2.88090402898857429640744064673, −1.78225191996541620238583767100, 0,
1.78225191996541620238583767100, 2.88090402898857429640744064673, 3.27394640911317126436954755060, 4.01857204132143657056562431371, 4.59262049442010086213140854358, 5.49902374106986148619709497199, 6.30669914534200722044857501360, 6.83704582334305772487332733025, 7.31693985110419627784309830236
