| L(s) = 1 | − 2.83·3-s + 2.22·5-s + 5.01·9-s − 1.51·11-s − 6.85·13-s − 6.28·15-s − 1.59·17-s + 2.36·19-s + 4.25·23-s − 0.0656·25-s − 5.70·27-s + 1.45·29-s − 2.09·31-s + 4.29·33-s + 7.96·37-s + 19.3·39-s + 41-s + 3.59·43-s + 11.1·45-s + 6.92·47-s + 4.51·51-s − 7.14·53-s − 3.36·55-s − 6.68·57-s + 3.14·59-s − 3.64·61-s − 15.2·65-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.993·5-s + 1.67·9-s − 0.456·11-s − 1.89·13-s − 1.62·15-s − 0.386·17-s + 0.542·19-s + 0.887·23-s − 0.0131·25-s − 1.09·27-s + 0.269·29-s − 0.375·31-s + 0.746·33-s + 1.30·37-s + 3.10·39-s + 0.156·41-s + 0.547·43-s + 1.66·45-s + 1.01·47-s + 0.632·51-s − 0.982·53-s − 0.453·55-s − 0.886·57-s + 0.409·59-s − 0.466·61-s − 1.88·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 6.85T + 13T^{2} \) |
| 17 | \( 1 + 1.59T + 17T^{2} \) |
| 19 | \( 1 - 2.36T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 7.96T + 37T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + 0.569T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 5.82T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.19896416139573537779728250489, −6.77691646245997816766442253032, −5.84947210953202969668824230653, −5.54141055229284275081483071602, −4.85099281617070702198467814621, −4.35499281433496888278411133812, −2.87466363594920051398846870203, −2.16431025994143863988007351344, −1.04782595444501526414712978071, 0,
1.04782595444501526414712978071, 2.16431025994143863988007351344, 2.87466363594920051398846870203, 4.35499281433496888278411133812, 4.85099281617070702198467814621, 5.54141055229284275081483071602, 5.84947210953202969668824230653, 6.77691646245997816766442253032, 7.19896416139573537779728250489
