| L(s) = 1 | + 2.66·2-s + 3.14·3-s + 5.12·4-s − 5-s + 8.39·6-s − 4.63·7-s + 8.32·8-s + 6.89·9-s − 2.66·10-s − 11-s + 16.1·12-s − 4.17·13-s − 12.3·14-s − 3.14·15-s + 11.9·16-s + 0.807·17-s + 18.3·18-s + 19-s − 5.12·20-s − 14.5·21-s − 2.66·22-s − 3.45·23-s + 26.1·24-s + 25-s − 11.1·26-s + 12.2·27-s − 23.7·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 1.81·3-s + 2.56·4-s − 0.447·5-s + 3.42·6-s − 1.75·7-s + 2.94·8-s + 2.29·9-s − 0.843·10-s − 0.301·11-s + 4.64·12-s − 1.15·13-s − 3.30·14-s − 0.812·15-s + 2.99·16-s + 0.195·17-s + 4.33·18-s + 0.229·19-s − 1.14·20-s − 3.17·21-s − 0.568·22-s − 0.719·23-s + 5.34·24-s + 0.200·25-s − 2.18·26-s + 2.35·27-s − 4.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.968248661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.968248661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 0.807T + 17T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 + 8.90T + 29T^{2} \) |
| 31 | \( 1 + 4.12T + 31T^{2} \) |
| 37 | \( 1 - 9.72T + 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 + 3.48T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.96T + 73T^{2} \) |
| 79 | \( 1 - 8.84T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−9.780057409104389092173968024100, −9.299727103787927855043791340090, −7.77888844911124180773704547477, −7.39181706862156995458153466729, −6.54211505270881833849768264284, −5.47272826171763790243580620176, −4.16104157941282961167082844341, −3.67708021815851850811037394565, −2.85748519854116744548190259408, −2.26993816986381150881854495186,
2.26993816986381150881854495186, 2.85748519854116744548190259408, 3.67708021815851850811037394565, 4.16104157941282961167082844341, 5.47272826171763790243580620176, 6.54211505270881833849768264284, 7.39181706862156995458153466729, 7.77888844911124180773704547477, 9.299727103787927855043791340090, 9.780057409104389092173968024100
