| L(s) = 1 | − 0.710·2-s − 3-s − 1.49·4-s − 5-s + 0.710·6-s − 4.35·7-s + 2.48·8-s + 9-s + 0.710·10-s + 4.62·11-s + 1.49·12-s + 1.23·13-s + 3.09·14-s + 15-s + 1.22·16-s − 5.04·17-s − 0.710·18-s − 2.56·19-s + 1.49·20-s + 4.35·21-s − 3.28·22-s − 2.48·24-s + 25-s − 0.874·26-s − 27-s + 6.51·28-s + 1.63·29-s + ⋯ |
| L(s) = 1 | − 0.502·2-s − 0.577·3-s − 0.747·4-s − 0.447·5-s + 0.290·6-s − 1.64·7-s + 0.878·8-s + 0.333·9-s + 0.224·10-s + 1.39·11-s + 0.431·12-s + 0.341·13-s + 0.827·14-s + 0.258·15-s + 0.306·16-s − 1.22·17-s − 0.167·18-s − 0.589·19-s + 0.334·20-s + 0.950·21-s − 0.700·22-s − 0.507·24-s + 0.200·25-s − 0.171·26-s − 0.192·27-s + 1.23·28-s + 0.302·29-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 | \( 1 + 0.710T + 2T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 + 6.01T + 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 + 0.607T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 - 0.225T + 89T^{2} \) |
| 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
−7.38849138645808772879821292432, −6.63920882723759647075754582012, −6.42427264911508440172402433545, −5.46516419958468185237177745430, −4.51801969270681220630562857560, −3.88569337981181919536829369244, −3.43511555498140267727198025121, −2.02839728591659128530180676757, −0.827924583754307188427829219111, 0,
0.827924583754307188427829219111, 2.02839728591659128530180676757, 3.43511555498140267727198025121, 3.88569337981181919536829369244, 4.51801969270681220630562857560, 5.46516419958468185237177745430, 6.42427264911508440172402433545, 6.63920882723759647075754582012, 7.38849138645808772879821292432
