| L(s) = 1 | − 2.10·3-s − 2.93·5-s + 1.42·9-s − 5.80·11-s − 1.53·13-s + 6.17·15-s + 5.69·17-s + 3.30·19-s − 2.90·23-s + 3.61·25-s + 3.31·27-s − 1.53·29-s − 6.66·31-s + 12.2·33-s − 11.3·37-s + 3.22·39-s + 41-s − 2.99·43-s − 4.17·45-s + 8.97·47-s − 11.9·51-s + 2.87·53-s + 17.0·55-s − 6.95·57-s + 9.75·59-s + 6.56·61-s + 4.50·65-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.31·5-s + 0.474·9-s − 1.75·11-s − 0.425·13-s + 1.59·15-s + 1.38·17-s + 0.758·19-s − 0.606·23-s + 0.723·25-s + 0.638·27-s − 0.285·29-s − 1.19·31-s + 2.12·33-s − 1.86·37-s + 0.516·39-s + 0.156·41-s − 0.457·43-s − 0.622·45-s + 1.30·47-s − 1.67·51-s + 0.394·53-s + 2.29·55-s − 0.920·57-s + 1.27·59-s + 0.840·61-s + 0.558·65-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 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 - 8.97T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 9.75T + 59T^{2} \) |
| 61 | \( 1 - 6.56T + 61T^{2} \) |
| 67 | \( 1 - 8.35T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 2.02T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 4.80T + 89T^{2} \) |
| 97 | \( 1 + 4.61T + 97T^{2} \) |
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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
−7.37341250306674637888200668489, −7.04623205995849084289594350201, −5.83734799096960490424129413833, −5.28325696581349242632909515115, −5.03375409255752728736349796556, −3.86111303031440100860828572906, −3.31370795946666199625549295590, −2.24605491516223882914466635387, −0.77006470517629869076991076817, 0,
0.77006470517629869076991076817, 2.24605491516223882914466635387, 3.31370795946666199625549295590, 3.86111303031440100860828572906, 5.03375409255752728736349796556, 5.28325696581349242632909515115, 5.83734799096960490424129413833, 7.04623205995849084289594350201, 7.37341250306674637888200668489
