L(s) = 1 | + 3-s − 3.32·5-s − 1.13·7-s + 9-s − 0.398·11-s + 2.39·13-s − 3.32·15-s + 7.10·17-s − 1.53·19-s − 1.13·21-s − 3.25·23-s + 6.04·25-s + 27-s − 3.93·29-s − 7.78·31-s − 0.398·33-s + 3.78·35-s + 1.25·37-s + 2.39·39-s − 7.50·41-s + 9.69·43-s − 3.32·45-s − 8.71·47-s − 5.70·49-s + 7.10·51-s + 10.7·53-s + 1.32·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.48·5-s − 0.430·7-s + 0.333·9-s − 0.120·11-s + 0.665·13-s − 0.858·15-s + 1.72·17-s − 0.352·19-s − 0.248·21-s − 0.679·23-s + 1.20·25-s + 0.192·27-s − 0.730·29-s − 1.39·31-s − 0.0693·33-s + 0.639·35-s + 0.206·37-s + 0.384·39-s − 1.17·41-s + 1.47·43-s − 0.495·45-s − 1.27·47-s − 0.814·49-s + 0.995·51-s + 1.47·53-s + 0.178·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 0.398T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.25T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 61 | \( 1 - 0.989T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 + 7.04T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.340675897521777457356740296536, −7.64765110918979118682210731637, −7.25820508138248960844403154624, −6.15226307420011633173504766031, −5.28493609143822908908787828110, −4.07581203651196201461264081858, −3.66552095785634142075693185452, −2.92011767899068907082842729392, −1.46238698901718465497944506666, 0,
1.46238698901718465497944506666, 2.92011767899068907082842729392, 3.66552095785634142075693185452, 4.07581203651196201461264081858, 5.28493609143822908908787828110, 6.15226307420011633173504766031, 7.25820508138248960844403154624, 7.64765110918979118682210731637, 8.340675897521777457356740296536