| L(s) = 1 | − 2-s + 4-s − 3.73·5-s − 2.73·7-s − 8-s + 3.73·10-s − 1.26·11-s + 2.73·14-s + 16-s + 5.73·17-s + 4.73·19-s − 3.73·20-s + 1.26·22-s − 4.19·23-s + 8.92·25-s − 2.73·28-s + 4.46·29-s + 1.46·31-s − 32-s − 5.73·34-s + 10.1·35-s + 3.53·37-s − 4.73·38-s + 3.73·40-s + 9.39·41-s − 9.66·43-s − 1.26·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.66·5-s − 1.03·7-s − 0.353·8-s + 1.18·10-s − 0.382·11-s + 0.730·14-s + 0.250·16-s + 1.39·17-s + 1.08·19-s − 0.834·20-s + 0.270·22-s − 0.874·23-s + 1.78·25-s − 0.516·28-s + 0.828·29-s + 0.262·31-s − 0.176·32-s − 0.983·34-s + 1.72·35-s + 0.581·37-s − 0.767·38-s + 0.590·40-s + 1.46·41-s − 1.47·43-s − 0.191·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 9.19T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.964148992169479860984990627056, −7.927896683684389523089182425908, −7.07704081481488914219961173888, −6.28269004821249121774789023850, −5.33078490083814939269586236629, −4.21904634530811705282532200219, −3.36429126814305742618972205917, −2.84453503914367121914553257580, −1.06324582863952725990020118708, 0,
1.06324582863952725990020118708, 2.84453503914367121914553257580, 3.36429126814305742618972205917, 4.21904634530811705282532200219, 5.33078490083814939269586236629, 6.28269004821249121774789023850, 7.07704081481488914219961173888, 7.927896683684389523089182425908, 7.964148992169479860984990627056
