| L(s) = 1 | − 3.33·3-s + 5-s + 2.42·7-s + 8.12·9-s + 3.26·11-s − 2.93·13-s − 3.33·15-s + 7.86·17-s − 5.45·19-s − 8.09·21-s + 3.39·23-s + 25-s − 17.0·27-s − 1.06·29-s + 9.81·31-s − 10.8·33-s + 2.42·35-s + 7.23·37-s + 9.79·39-s − 1.82·41-s + 2.53·43-s + 8.12·45-s + 8.13·47-s − 1.11·49-s − 26.2·51-s + 7.28·53-s + 3.26·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 0.447·5-s + 0.917·7-s + 2.70·9-s + 0.983·11-s − 0.814·13-s − 0.861·15-s + 1.90·17-s − 1.25·19-s − 1.76·21-s + 0.707·23-s + 0.200·25-s − 3.28·27-s − 0.198·29-s + 1.76·31-s − 1.89·33-s + 0.410·35-s + 1.19·37-s + 1.56·39-s − 0.284·41-s + 0.387·43-s + 1.21·45-s + 1.18·47-s − 0.158·49-s − 3.67·51-s + 1.00·53-s + 0.439·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.603582361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603582361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 - 9.81T + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 - 8.13T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.49T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 - 4.71T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63587852670804640546470419140, −6.92369362366662891479538532274, −6.35686622464356649208879921205, −5.70158777607742843103446035070, −5.15561541970562448737392474769, −4.53553871954111188101603665182, −3.90224564576750251321954634618, −2.43344502629801704375780748362, −1.33807305755518299639080487804, −0.815912575519894519518246887155,
0.815912575519894519518246887155, 1.33807305755518299639080487804, 2.43344502629801704375780748362, 3.90224564576750251321954634618, 4.53553871954111188101603665182, 5.15561541970562448737392474769, 5.70158777607742843103446035070, 6.35686622464356649208879921205, 6.92369362366662891479538532274, 7.63587852670804640546470419140
