| L(s) = 1 | + 2.17·2-s + 2.53·3-s + 2.73·4-s − 5-s + 5.51·6-s − 4.33·7-s + 1.59·8-s + 3.42·9-s − 2.17·10-s − 5.75·11-s + 6.92·12-s − 2.80·13-s − 9.42·14-s − 2.53·15-s − 1.99·16-s − 2.17·17-s + 7.44·18-s + 4.90·19-s − 2.73·20-s − 10.9·21-s − 12.5·22-s + 2.10·23-s + 4.04·24-s + 25-s − 6.09·26-s + 1.06·27-s − 11.8·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.46·3-s + 1.36·4-s − 0.447·5-s + 2.25·6-s − 1.63·7-s + 0.564·8-s + 1.14·9-s − 0.688·10-s − 1.73·11-s + 1.99·12-s − 0.776·13-s − 2.51·14-s − 0.654·15-s − 0.498·16-s − 0.527·17-s + 1.75·18-s + 1.12·19-s − 0.611·20-s − 2.39·21-s − 2.66·22-s + 0.439·23-s + 0.826·24-s + 0.200·25-s − 1.19·26-s + 0.205·27-s − 2.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 + 4.33T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 37 | \( 1 - 8.20T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 + 0.841T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 + 8.86T + 97T^{2} \) |
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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
−7.80665360375075774142246551920, −7.25825407227941458230968841298, −6.61834771298350299945511309191, −5.55933320246981021990080105849, −4.98320858796159843332052016136, −3.96293450285986678393070342984, −3.31455027321194558253836793753, −2.83194943197546193211809903873, −2.31126936739112584814232903178, 0,
2.31126936739112584814232903178, 2.83194943197546193211809903873, 3.31455027321194558253836793753, 3.96293450285986678393070342984, 4.98320858796159843332052016136, 5.55933320246981021990080105849, 6.61834771298350299945511309191, 7.25825407227941458230968841298, 7.80665360375075774142246551920
