| L(s) = 1 | − 0.167·3-s − 5-s + 2.31·7-s − 2.97·9-s + 2.88·11-s + 4.00·13-s + 0.167·15-s + 4.29·17-s − 0.543·19-s − 0.388·21-s − 1.32·23-s + 25-s + 1.00·27-s − 6.26·29-s + 0.454·31-s − 0.485·33-s − 2.31·35-s − 1.85·37-s − 0.673·39-s + 8.59·41-s − 7.75·43-s + 2.97·45-s + 12.0·47-s − 1.66·49-s − 0.721·51-s + 9.04·53-s − 2.88·55-s + ⋯ |
| L(s) = 1 | − 0.0969·3-s − 0.447·5-s + 0.873·7-s − 0.990·9-s + 0.870·11-s + 1.11·13-s + 0.0433·15-s + 1.04·17-s − 0.124·19-s − 0.0846·21-s − 0.275·23-s + 0.200·25-s + 0.193·27-s − 1.16·29-s + 0.0816·31-s − 0.0844·33-s − 0.390·35-s − 0.305·37-s − 0.107·39-s + 1.34·41-s − 1.18·43-s + 0.443·45-s + 1.76·47-s − 0.237·49-s − 0.100·51-s + 1.24·53-s − 0.389·55-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
| \(L(1)\) |
\(\approx\) |
\(2.134515783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134515783\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.167T + 3T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 0.543T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 - 0.454T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 9.04T + 53T^{2} \) |
| 59 | \( 1 - 3.28T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 3.48T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 3.13T + 97T^{2} \) |
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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
−8.030001299532674593612361974967, −7.18762120872350155921841402366, −6.41230893272790884709238763165, −5.63783827245945618142542052950, −5.23641509796682650560107569268, −4.01977464465602855952739459143, −3.75883402439550184458607839459, −2.70748747223738032647428371091, −1.64359273612466254996747724843, −0.76764166927982448758389033510,
0.76764166927982448758389033510, 1.64359273612466254996747724843, 2.70748747223738032647428371091, 3.75883402439550184458607839459, 4.01977464465602855952739459143, 5.23641509796682650560107569268, 5.63783827245945618142542052950, 6.41230893272790884709238763165, 7.18762120872350155921841402366, 8.030001299532674593612361974967
