| L(s) = 1 | + 2.06·3-s − 2.59·5-s − 3.89·7-s + 1.25·9-s − 0.755·11-s + 5.27·13-s − 5.34·15-s − 17-s − 7.19·19-s − 8.04·21-s + 5.60·23-s + 1.72·25-s − 3.59·27-s − 1.75·29-s − 0.233·31-s − 1.55·33-s + 10.1·35-s + 0.658·37-s + 10.8·39-s + 1.81·41-s + 4.61·43-s − 3.25·45-s − 7.68·47-s + 8.19·49-s − 2.06·51-s − 11.0·53-s + 1.95·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s − 1.15·5-s − 1.47·7-s + 0.419·9-s − 0.227·11-s + 1.46·13-s − 1.38·15-s − 0.242·17-s − 1.65·19-s − 1.75·21-s + 1.16·23-s + 0.344·25-s − 0.691·27-s − 0.325·29-s − 0.0419·31-s − 0.271·33-s + 1.70·35-s + 0.108·37-s + 1.74·39-s + 0.284·41-s + 0.704·43-s − 0.485·45-s − 1.12·47-s + 1.17·49-s − 0.288·51-s − 1.51·53-s + 0.263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449896532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449896532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 + 0.755T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 + 0.233T + 31T^{2} \) |
| 37 | \( 1 - 0.658T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 6.31T + 71T^{2} \) |
| 73 | \( 1 - 0.789T + 73T^{2} \) |
| 79 | \( 1 - 4.02T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 4.33T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−8.015593451196673908654311845345, −7.26306909931153173544953029985, −6.46694523999134896147386026035, −6.06751764084716685620486987891, −4.77852541395921849602438091060, −3.86958611456761446166605756453, −3.52950935294384195582080432376, −2.93943897851858890683861686869, −2.00115394022507855347609522644, −0.53422716380063101400709488865,
0.53422716380063101400709488865, 2.00115394022507855347609522644, 2.93943897851858890683861686869, 3.52950935294384195582080432376, 3.86958611456761446166605756453, 4.77852541395921849602438091060, 6.06751764084716685620486987891, 6.46694523999134896147386026035, 7.26306909931153173544953029985, 8.015593451196673908654311845345
