| L(s) = 1 | − 3-s − 1.10·5-s − 3.58·7-s + 9-s + 5.82·11-s − 1.93·13-s + 1.10·15-s + 2.35·17-s − 8.34·19-s + 3.58·21-s + 8.17·23-s − 3.77·25-s − 27-s − 1.11·29-s + 6.08·31-s − 5.82·33-s + 3.97·35-s + 1.22·37-s + 1.93·39-s − 0.437·41-s − 5.51·43-s − 1.10·45-s + 1.26·47-s + 5.85·49-s − 2.35·51-s + 2.37·53-s − 6.45·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.495·5-s − 1.35·7-s + 0.333·9-s + 1.75·11-s − 0.536·13-s + 0.286·15-s + 0.570·17-s − 1.91·19-s + 0.782·21-s + 1.70·23-s − 0.754·25-s − 0.192·27-s − 0.206·29-s + 1.09·31-s − 1.01·33-s + 0.671·35-s + 0.201·37-s + 0.309·39-s − 0.0682·41-s − 0.840·43-s − 0.165·45-s + 0.184·47-s + 0.837·49-s − 0.329·51-s + 0.326·53-s − 0.870·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 0.437T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 2.91T + 59T^{2} \) |
| 61 | \( 1 + 2.63T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.107859251038515606527845180027, −6.99594147548958579794594578648, −6.64315126698826802056713373144, −6.11794713096392208710160014263, −5.05899993564544597437343302737, −4.11033611078403321205637257837, −3.63598626954446461835768797022, −2.55841507797118741975635694092, −1.17828345971793643390830037949, 0,
1.17828345971793643390830037949, 2.55841507797118741975635694092, 3.63598626954446461835768797022, 4.11033611078403321205637257837, 5.05899993564544597437343302737, 6.11794713096392208710160014263, 6.64315126698826802056713373144, 6.99594147548958579794594578648, 8.107859251038515606527845180027
