| L(s) = 1 | − 2.46·2-s − 3-s + 4.07·4-s − 5-s + 2.46·6-s − 1.15·7-s − 5.10·8-s + 9-s + 2.46·10-s + 4.09·11-s − 4.07·12-s − 2.47·13-s + 2.83·14-s + 15-s + 4.44·16-s − 2.96·17-s − 2.46·18-s + 0.682·19-s − 4.07·20-s + 1.15·21-s − 10.0·22-s + 5.10·24-s + 25-s + 6.08·26-s − 27-s − 4.68·28-s − 0.0686·29-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.03·4-s − 0.447·5-s + 1.00·6-s − 0.434·7-s − 1.80·8-s + 0.333·9-s + 0.779·10-s + 1.23·11-s − 1.17·12-s − 0.685·13-s + 0.757·14-s + 0.258·15-s + 1.11·16-s − 0.718·17-s − 0.580·18-s + 0.156·19-s − 0.910·20-s + 0.251·21-s − 2.15·22-s + 1.04·24-s + 0.200·25-s + 1.19·26-s − 0.192·27-s − 0.885·28-s − 0.0127·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 0.682T + 19T^{2} \) |
| 29 | \( 1 + 0.0686T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 4.35T + 61T^{2} \) |
| 67 | \( 1 + 3.24T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 5.50T + 73T^{2} \) |
| 79 | \( 1 + 0.165T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.62953846770193087486443865015, −6.83375943011653014179681184033, −6.57237309872005636763540143738, −5.73357681313496001742697178592, −4.62890334821213145892613698850, −3.86493864249819928909932143684, −2.78981984808353749677515974332, −1.85087053866434054195016408462, −0.918187160900491144939478693317, 0,
0.918187160900491144939478693317, 1.85087053866434054195016408462, 2.78981984808353749677515974332, 3.86493864249819928909932143684, 4.62890334821213145892613698850, 5.73357681313496001742697178592, 6.57237309872005636763540143738, 6.83375943011653014179681184033, 7.62953846770193087486443865015
