| L(s) = 1 | + 2-s + 4-s + 3.76·5-s + 7-s + 8-s + 3.76·10-s + 5.76·11-s − 4.26·13-s + 14-s + 16-s + 4.26·17-s − 8.03·19-s + 3.76·20-s + 5.76·22-s + 23-s + 9.18·25-s − 4.26·26-s + 28-s + 2·29-s + 0.0811·31-s + 32-s + 4.26·34-s + 3.76·35-s + 0.579·37-s − 8.03·38-s + 3.76·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.68·5-s + 0.377·7-s + 0.353·8-s + 1.19·10-s + 1.73·11-s − 1.18·13-s + 0.267·14-s + 0.250·16-s + 1.03·17-s − 1.84·19-s + 0.842·20-s + 1.22·22-s + 0.208·23-s + 1.83·25-s − 0.836·26-s + 0.188·28-s + 0.371·29-s + 0.0145·31-s + 0.176·32-s + 0.731·34-s + 0.636·35-s + 0.0953·37-s − 1.30·38-s + 0.595·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.479901128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.479901128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.76T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 8.03T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 0.0811T + 31T^{2} \) |
| 37 | \( 1 - 0.579T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 - 3.42T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−9.082009171834805809178528253091, −7.926310380406619456915941907103, −6.92845051190911000828080973050, −6.31274070638982889649220669698, −5.80424936515362052674980389513, −4.86149267969707143959088249959, −4.23601720073400727847612449541, −3.01071144213378302324550733193, −2.07281686113120558942125832060, −1.37350238077133991692520456472,
1.37350238077133991692520456472, 2.07281686113120558942125832060, 3.01071144213378302324550733193, 4.23601720073400727847612449541, 4.86149267969707143959088249959, 5.80424936515362052674980389513, 6.31274070638982889649220669698, 6.92845051190911000828080973050, 7.926310380406619456915941907103, 9.082009171834805809178528253091
