| L(s) = 1 | − 1.26·2-s − 3-s − 0.404·4-s − 5-s + 1.26·6-s − 1.43·7-s + 3.03·8-s + 9-s + 1.26·10-s − 0.845·11-s + 0.404·12-s − 0.982·13-s + 1.81·14-s + 15-s − 3.02·16-s + 2.92·17-s − 1.26·18-s + 4.80·19-s + 0.404·20-s + 1.43·21-s + 1.06·22-s − 4.19·23-s − 3.03·24-s + 25-s + 1.24·26-s − 27-s + 0.581·28-s + ⋯ |
| L(s) = 1 | − 0.893·2-s − 0.577·3-s − 0.202·4-s − 0.447·5-s + 0.515·6-s − 0.542·7-s + 1.07·8-s + 0.333·9-s + 0.399·10-s − 0.254·11-s + 0.116·12-s − 0.272·13-s + 0.484·14-s + 0.258·15-s − 0.756·16-s + 0.710·17-s − 0.297·18-s + 1.10·19-s + 0.0904·20-s + 0.313·21-s + 0.227·22-s − 0.873·23-s − 0.619·24-s + 0.200·25-s + 0.243·26-s − 0.192·27-s + 0.109·28-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 | \( 1 + 1.26T + 2T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 0.845T + 11T^{2} \) |
| 13 | \( 1 + 0.982T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 - 4.01T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 0.916T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−7.85119177644040400104876495125, −7.13157024314582318596892776010, −6.55297818876754440443470262271, −5.35148881660382129215742020743, −5.11334281240141461474972412624, −3.94412040455905633817951342309, −3.37666124429344492577001316605, −2.02680895108922603456062511594, −0.925606825499219918205766757134, 0,
0.925606825499219918205766757134, 2.02680895108922603456062511594, 3.37666124429344492577001316605, 3.94412040455905633817951342309, 5.11334281240141461474972412624, 5.35148881660382129215742020743, 6.55297818876754440443470262271, 7.13157024314582318596892776010, 7.85119177644040400104876495125
