| L(s) = 1 | − 2.15·2-s + 1.54·3-s + 2.66·4-s + 5-s − 3.33·6-s + 2.48·7-s − 1.42·8-s − 0.609·9-s − 2.15·10-s − 11-s + 4.11·12-s + 2.62·13-s − 5.35·14-s + 1.54·15-s − 2.24·16-s − 6.48·17-s + 1.31·18-s − 3.23·19-s + 2.66·20-s + 3.83·21-s + 2.15·22-s − 1.58·23-s − 2.20·24-s + 25-s − 5.67·26-s − 5.58·27-s + 6.60·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.892·3-s + 1.33·4-s + 0.447·5-s − 1.36·6-s + 0.938·7-s − 0.504·8-s − 0.203·9-s − 0.682·10-s − 0.301·11-s + 1.18·12-s + 0.729·13-s − 1.43·14-s + 0.399·15-s − 0.560·16-s − 1.57·17-s + 0.310·18-s − 0.742·19-s + 0.594·20-s + 0.837·21-s + 0.460·22-s − 0.330·23-s − 0.450·24-s + 0.200·25-s − 1.11·26-s − 1.07·27-s + 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 - 1.54T + 3T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 8.57T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 + 5.30T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 5.21T + 97T^{2} \) |
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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
−8.229447468126915609135143117953, −7.900629207478762777442029255810, −6.78769090835133355340581921785, −6.26335343616307461955219534803, −5.03425264744430935880714051864, −4.23043267023430285116500179747, −2.97724501513942878014865561727, −2.07985144980491210156849627648, −1.56854283936847364194962927168, 0,
1.56854283936847364194962927168, 2.07985144980491210156849627648, 2.97724501513942878014865561727, 4.23043267023430285116500179747, 5.03425264744430935880714051864, 6.26335343616307461955219534803, 6.78769090835133355340581921785, 7.900629207478762777442029255810, 8.229447468126915609135143117953
