| L(s) = 1 | + 1.52·2-s + 0.316·4-s − 1.48·7-s − 2.56·8-s − 0.730·11-s + 2.32·13-s − 2.25·14-s − 4.53·16-s + 6.58·17-s − 19-s − 1.11·22-s − 2.31·23-s + 3.54·26-s − 0.467·28-s − 5.60·29-s + 3.84·31-s − 1.77·32-s + 10.0·34-s − 1.67·37-s − 1.52·38-s − 3.29·41-s − 7.69·43-s − 0.230·44-s − 3.51·46-s − 3.54·47-s − 4.80·49-s + 0.735·52-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.158·4-s − 0.559·7-s − 0.906·8-s − 0.220·11-s + 0.645·13-s − 0.602·14-s − 1.13·16-s + 1.59·17-s − 0.229·19-s − 0.237·22-s − 0.482·23-s + 0.694·26-s − 0.0884·28-s − 1.04·29-s + 0.691·31-s − 0.313·32-s + 1.71·34-s − 0.274·37-s − 0.246·38-s − 0.514·41-s − 1.17·43-s − 0.0348·44-s − 0.518·46-s − 0.516·47-s − 0.686·49-s + 0.102·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 + 0.730T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 + 0.480T + 53T^{2} \) |
| 59 | \( 1 - 0.249T + 59T^{2} \) |
| 61 | \( 1 + 3.73T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 0.249T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 9.54T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 2.32T + 97T^{2} \) |
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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
−8.045228995625568084327063940038, −7.11206749972782130132978587655, −6.25850873118319020259041029530, −5.75563028505997434781780728172, −5.06620131832482238110776888285, −4.18104730689263100273957451293, −3.42023378986795535385894195790, −2.93149998006762743486440943232, −1.54784722714706040952869705058, 0,
1.54784722714706040952869705058, 2.93149998006762743486440943232, 3.42023378986795535385894195790, 4.18104730689263100273957451293, 5.06620131832482238110776888285, 5.75563028505997434781780728172, 6.25850873118319020259041029530, 7.11206749972782130132978587655, 8.045228995625568084327063940038
