| L(s) = 1 | − 1.32·2-s − 0.254·4-s − 3.90·7-s + 2.97·8-s − 2.89·11-s + 0.529·13-s + 5.15·14-s − 3.42·16-s − 3.91·17-s + 0.525·19-s + 3.82·22-s − 4.99·23-s − 0.699·26-s + 0.991·28-s + 7.35·29-s + 0.179·31-s − 1.42·32-s + 5.17·34-s + 37-s − 0.693·38-s + 0.810·41-s + 9.21·43-s + 0.734·44-s + 6.60·46-s + 10.9·47-s + 8.21·49-s − 0.134·52-s + ⋯ |
| L(s) = 1 | − 0.934·2-s − 0.127·4-s − 1.47·7-s + 1.05·8-s − 0.871·11-s + 0.146·13-s + 1.37·14-s − 0.856·16-s − 0.950·17-s + 0.120·19-s + 0.814·22-s − 1.04·23-s − 0.137·26-s + 0.187·28-s + 1.36·29-s + 0.0323·31-s − 0.252·32-s + 0.888·34-s + 0.164·37-s − 0.112·38-s + 0.126·41-s + 1.40·43-s + 0.110·44-s + 0.973·46-s + 1.59·47-s + 1.17·49-s − 0.0186·52-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 | \( 1 + 1.32T + 2T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 0.529T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 - 0.525T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 31 | \( 1 - 0.179T + 31T^{2} \) |
| 41 | \( 1 - 0.810T + 41T^{2} \) |
| 43 | \( 1 - 9.21T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 3.78T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.52822009354816843635930770784, −6.93894265612526947344437429323, −6.19071445258975890642504501665, −5.53357069017785977515195236874, −4.49538082531412696581055576991, −3.95446229724206000151043661850, −2.88601425709128307771409979294, −2.21988059157728645190192191222, −0.852872519909622705906004566291, 0,
0.852872519909622705906004566291, 2.21988059157728645190192191222, 2.88601425709128307771409979294, 3.95446229724206000151043661850, 4.49538082531412696581055576991, 5.53357069017785977515195236874, 6.19071445258975890642504501665, 6.93894265612526947344437429323, 7.52822009354816843635930770784
