| L(s) = 1 | − 1.50·2-s − 3-s + 0.261·4-s − 5-s + 1.50·6-s + 2.03·7-s + 2.61·8-s + 9-s + 1.50·10-s − 4.89·11-s − 0.261·12-s − 6.58·13-s − 3.05·14-s + 15-s − 4.45·16-s − 5.34·17-s − 1.50·18-s + 7.00·19-s − 0.261·20-s − 2.03·21-s + 7.35·22-s − 0.703·23-s − 2.61·24-s + 25-s + 9.89·26-s − 27-s + 0.531·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 0.130·4-s − 0.447·5-s + 0.613·6-s + 0.768·7-s + 0.924·8-s + 0.333·9-s + 0.475·10-s − 1.47·11-s − 0.0754·12-s − 1.82·13-s − 0.817·14-s + 0.258·15-s − 1.11·16-s − 1.29·17-s − 0.354·18-s + 1.60·19-s − 0.0584·20-s − 0.443·21-s + 1.56·22-s − 0.146·23-s − 0.533·24-s + 0.200·25-s + 1.94·26-s − 0.192·27-s + 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6.58T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 23 | \( 1 + 0.703T + 23T^{2} \) |
| 29 | \( 1 - 9.26T + 29T^{2} \) |
| 31 | \( 1 - 7.74T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 + 7.06T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.61T + 97T^{2} \) |
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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
−7.911471384576927968430120421832, −7.28255076261248628834228997552, −6.61947885187604157921446935831, −5.32868638893573088152203873124, −4.76171544616221257975691273942, −4.53445894404213078105780237553, −2.91672871226931311400204174398, −2.15833579718443998125127582815, −0.906533425422992786341244371441, 0,
0.906533425422992786341244371441, 2.15833579718443998125127582815, 2.91672871226931311400204174398, 4.53445894404213078105780237553, 4.76171544616221257975691273942, 5.32868638893573088152203873124, 6.61947885187604157921446935831, 7.28255076261248628834228997552, 7.911471384576927968430120421832
