| L(s) = 1 | − 3-s + 5-s + 0.778·7-s + 9-s − 2.24·11-s + 4.24·13-s − 15-s − 0.529·17-s − 3.30·19-s − 0.778·21-s − 23-s + 25-s − 27-s − 2.41·29-s − 3.71·31-s + 2.24·33-s + 0.778·35-s + 2.77·37-s − 4.24·39-s − 11.6·41-s − 6.94·43-s + 45-s − 1.19·47-s − 6.39·49-s + 0.529·51-s − 1.58·53-s − 2.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.294·7-s + 0.333·9-s − 0.678·11-s + 1.17·13-s − 0.258·15-s − 0.128·17-s − 0.758·19-s − 0.169·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.447·29-s − 0.668·31-s + 0.391·33-s + 0.131·35-s + 0.456·37-s − 0.680·39-s − 1.81·41-s − 1.05·43-s + 0.149·45-s − 0.173·47-s − 0.913·49-s + 0.0741·51-s − 0.218·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 0.778T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 0.529T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.048416828083141898141868460278, −6.75904527177945357086711005543, −6.50217144066444844475024390964, −5.50293879356424681670806400867, −5.12705907743973796983922289746, −4.12953382454728826607417282560, −3.34695822995481466213377612155, −2.17079265004713863937461432270, −1.38706060858033946867150350593, 0,
1.38706060858033946867150350593, 2.17079265004713863937461432270, 3.34695822995481466213377612155, 4.12953382454728826607417282560, 5.12705907743973796983922289746, 5.50293879356424681670806400867, 6.50217144066444844475024390964, 6.75904527177945357086711005543, 8.048416828083141898141868460278
