| L(s) = 1 | + 3.38·3-s − 5-s + 4.25·7-s + 8.43·9-s − 5.65·11-s − 5.75·13-s − 3.38·15-s + 4.88·17-s + 1.98·19-s + 14.4·21-s + 3.47·23-s + 25-s + 18.3·27-s + 5.21·29-s + 9.46·31-s − 19.1·33-s − 4.25·35-s − 9.00·37-s − 19.4·39-s + 7.63·41-s − 0.0632·43-s − 8.43·45-s − 4.49·47-s + 11.1·49-s + 16.5·51-s + 3.48·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 0.447·5-s + 1.60·7-s + 2.81·9-s − 1.70·11-s − 1.59·13-s − 0.873·15-s + 1.18·17-s + 0.455·19-s + 3.14·21-s + 0.723·23-s + 0.200·25-s + 3.53·27-s + 0.968·29-s + 1.69·31-s − 3.32·33-s − 0.719·35-s − 1.48·37-s − 3.11·39-s + 1.19·41-s − 0.00963·43-s − 1.25·45-s − 0.655·47-s + 1.59·49-s + 2.31·51-s + 0.478·53-s + 0.762·55-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
| \(L(1)\) |
\(\approx\) |
\(4.920496674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.920496674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.38T + 3T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 + 0.0632T + 43T^{2} \) |
| 47 | \( 1 + 4.49T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 3.10T + 79T^{2} \) |
| 83 | \( 1 + 1.90T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
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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
−7.903331142369029850323444881334, −7.52667295964694599224637387935, −7.00271314231116820235955451814, −5.38114496958723957663457175510, −4.78178915853242474012218543145, −4.41205456689415566130067408791, −3.13781973177255276921644298339, −2.78314532379246355305532872623, −2.04164092084120515409772563943, −1.05141516960072903815054132660,
1.05141516960072903815054132660, 2.04164092084120515409772563943, 2.78314532379246355305532872623, 3.13781973177255276921644298339, 4.41205456689415566130067408791, 4.78178915853242474012218543145, 5.38114496958723957663457175510, 7.00271314231116820235955451814, 7.52667295964694599224637387935, 7.903331142369029850323444881334
