| L(s) = 1 | + 0.747·2-s − 1.44·4-s − 3.28·7-s − 2.57·8-s + 1.14·11-s + 2.14·13-s − 2.45·14-s + 0.958·16-s + 6.16·17-s − 7.20·19-s + 0.853·22-s − 0.227·23-s + 1.60·26-s + 4.73·28-s + 29-s + 1.59·31-s + 5.86·32-s + 4.60·34-s + 0.690·37-s − 5.38·38-s + 9.55·41-s + 0.739·43-s − 1.64·44-s − 0.170·46-s − 3.52·47-s + 3.80·49-s − 3.09·52-s + ⋯ |
| L(s) = 1 | + 0.528·2-s − 0.720·4-s − 1.24·7-s − 0.909·8-s + 0.344·11-s + 0.595·13-s − 0.656·14-s + 0.239·16-s + 1.49·17-s − 1.65·19-s + 0.181·22-s − 0.0474·23-s + 0.314·26-s + 0.895·28-s + 0.185·29-s + 0.285·31-s + 1.03·32-s + 0.790·34-s + 0.113·37-s − 0.873·38-s + 1.49·41-s + 0.112·43-s − 0.248·44-s − 0.0250·46-s − 0.514·47-s + 0.544·49-s − 0.428·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.747T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 + 0.227T + 23T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 - 0.690T + 37T^{2} \) |
| 41 | \( 1 - 9.55T + 41T^{2} \) |
| 43 | \( 1 - 0.739T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 0.756T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 + 0.836T + 61T^{2} \) |
| 67 | \( 1 + 6.00T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 + 0.620T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−7.66355292533562620356126899397, −6.73437690482299884612221934611, −6.02068729912820264752844057604, −5.74714510830185429567556144276, −4.61660711004747213935838485981, −3.98802193718855203790360650028, −3.36303556407738172638295172926, −2.62820325589259683832266208545, −1.15571206242245060112864506073, 0,
1.15571206242245060112864506073, 2.62820325589259683832266208545, 3.36303556407738172638295172926, 3.98802193718855203790360650028, 4.61660711004747213935838485981, 5.74714510830185429567556144276, 6.02068729912820264752844057604, 6.73437690482299884612221934611, 7.66355292533562620356126899397
