| L(s) = 1 | − 3.20·3-s + 1.15·5-s + 3.99·7-s + 7.26·9-s + 3.92·11-s − 1.69·13-s − 3.70·15-s + 0.837·17-s − 1.86·19-s − 12.7·21-s − 1.87·23-s − 3.66·25-s − 13.6·27-s − 10.1·29-s − 9.22·31-s − 12.5·33-s + 4.61·35-s − 4.58·37-s + 5.43·39-s + 2.89·41-s + 7.59·43-s + 8.39·45-s − 7.88·47-s + 8.93·49-s − 2.68·51-s − 11.2·53-s + 4.53·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 0.516·5-s + 1.50·7-s + 2.42·9-s + 1.18·11-s − 0.470·13-s − 0.955·15-s + 0.203·17-s − 0.428·19-s − 2.79·21-s − 0.390·23-s − 0.732·25-s − 2.62·27-s − 1.87·29-s − 1.65·31-s − 2.18·33-s + 0.779·35-s − 0.753·37-s + 0.869·39-s + 0.452·41-s + 1.15·43-s + 1.25·45-s − 1.15·47-s + 1.27·49-s − 0.375·51-s − 1.54·53-s + 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 0.837T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 7.88T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.73877716840451381206119028133, −7.32184705405457038453812028802, −6.38633416605917768915372170397, −5.75534033668127187815201835912, −5.26272219547168523336346336745, −4.47634182155434909577028678410, −3.84822977270839381840951715935, −1.81591135089856087636516429464, −1.51994063905283880210585474241, 0,
1.51994063905283880210585474241, 1.81591135089856087636516429464, 3.84822977270839381840951715935, 4.47634182155434909577028678410, 5.26272219547168523336346336745, 5.75534033668127187815201835912, 6.38633416605917768915372170397, 7.32184705405457038453812028802, 7.73877716840451381206119028133
