| L(s) = 1 | + 3-s + 5-s − 1.14·7-s + 9-s − 4.68·11-s − 0.853·13-s + 15-s − 4·17-s + 7.66·19-s − 1.14·21-s + 4.29·23-s + 25-s + 27-s − 3.43·29-s − 31-s − 4.68·33-s − 1.14·35-s − 7.14·37-s − 0.853·39-s − 4.29·41-s + 9.37·43-s + 45-s − 2.68·47-s − 5.68·49-s − 4·51-s − 6.39·53-s − 4.68·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.433·7-s + 0.333·9-s − 1.41·11-s − 0.236·13-s + 0.258·15-s − 0.970·17-s + 1.75·19-s − 0.250·21-s + 0.895·23-s + 0.200·25-s + 0.192·27-s − 0.638·29-s − 0.179·31-s − 0.815·33-s − 0.193·35-s − 1.17·37-s − 0.136·39-s − 0.670·41-s + 1.42·43-s + 0.149·45-s − 0.391·47-s − 0.812·49-s − 0.560·51-s − 0.878·53-s − 0.631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + 4.68T + 11T^{2} \) |
| 13 | \( 1 + 0.853T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 37 | \( 1 + 7.14T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 6.39T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 3.43T + 67T^{2} \) |
| 71 | \( 1 - 9.20T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 + 0.393T + 79T^{2} \) |
| 83 | \( 1 - 6.68T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.49507441716153917414388231233, −7.03721004681474078045603723002, −6.18530430715905638339535593053, −5.21879128155046309154367468507, −4.97638346173686068823293772988, −3.75047660677088881442765611862, −2.99204036940605748888541398777, −2.44590549061611524208324639635, −1.41018763255481810499867960162, 0,
1.41018763255481810499867960162, 2.44590549061611524208324639635, 2.99204036940605748888541398777, 3.75047660677088881442765611862, 4.97638346173686068823293772988, 5.21879128155046309154367468507, 6.18530430715905638339535593053, 7.03721004681474078045603723002, 7.49507441716153917414388231233
