| L(s) = 1 | − 0.214·2-s − 1.95·4-s − 4.41·7-s + 0.848·8-s − 5.57·11-s − 0.780·13-s + 0.948·14-s + 3.72·16-s + 4.10·17-s − 4.94·19-s + 1.19·22-s + 7.70·23-s + 0.167·26-s + 8.63·28-s + 1.53·29-s − 2.52·31-s − 2.49·32-s − 0.881·34-s − 37-s + 1.06·38-s + 11.2·41-s + 2.78·43-s + 10.8·44-s − 1.65·46-s − 2.53·47-s + 12.5·49-s + 1.52·52-s + ⋯ |
| L(s) = 1 | − 0.151·2-s − 0.976·4-s − 1.67·7-s + 0.300·8-s − 1.68·11-s − 0.216·13-s + 0.253·14-s + 0.931·16-s + 0.996·17-s − 1.13·19-s + 0.255·22-s + 1.60·23-s + 0.0328·26-s + 1.63·28-s + 0.285·29-s − 0.453·31-s − 0.441·32-s − 0.151·34-s − 0.164·37-s + 0.172·38-s + 1.75·41-s + 0.424·43-s + 1.64·44-s − 0.243·46-s − 0.370·47-s + 1.78·49-s + 0.211·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.214T + 2T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 0.780T + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 0.343T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.50692618279247943642790739336, −6.85951070745922026466877334028, −5.92891222808443822111658588234, −5.42289030124469876925561791740, −4.69484549300959493855628829961, −3.81546704575330686066293421159, −3.07958367846854747025056693976, −2.48043951330510396741594381439, −0.852481334568256764369237402586, 0,
0.852481334568256764369237402586, 2.48043951330510396741594381439, 3.07958367846854747025056693976, 3.81546704575330686066293421159, 4.69484549300959493855628829961, 5.42289030124469876925561791740, 5.92891222808443822111658588234, 6.85951070745922026466877334028, 7.50692618279247943642790739336
