| L(s) = 1 | + 1.49·3-s + 1.55·5-s − 0.767·9-s − 2.64·11-s − 2.36·13-s + 2.32·15-s − 17-s + 3.07·19-s + 1.26·23-s − 2.58·25-s − 5.62·27-s + 3.70·29-s − 8.82·31-s − 3.94·33-s − 7.10·37-s − 3.53·39-s + 5.19·41-s + 5.83·43-s − 1.19·45-s + 2.36·47-s − 1.49·51-s + 4.45·53-s − 4.10·55-s + 4.59·57-s − 4.78·59-s − 9.63·61-s − 3.67·65-s + ⋯ |
| L(s) = 1 | + 0.862·3-s + 0.694·5-s − 0.255·9-s − 0.796·11-s − 0.656·13-s + 0.599·15-s − 0.242·17-s + 0.706·19-s + 0.263·23-s − 0.517·25-s − 1.08·27-s + 0.687·29-s − 1.58·31-s − 0.686·33-s − 1.16·37-s − 0.566·39-s + 0.811·41-s + 0.889·43-s − 0.177·45-s + 0.344·47-s − 0.209·51-s + 0.612·53-s − 0.553·55-s + 0.609·57-s − 0.622·59-s − 1.23·61-s − 0.456·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 7.10T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 4.45T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 9.63T + 61T^{2} \) |
| 67 | \( 1 + 8.78T + 67T^{2} \) |
| 71 | \( 1 - 2.58T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2.57T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.47194377080697428615717467585, −7.33099138824343281718675482683, −6.06287431340825476662839498481, −5.57042614644503521028691688059, −4.84564013692593615220452440900, −3.85266391878274018026939905146, −2.95219300692453178298894419696, −2.44409693399062889598354321935, −1.58126280198489359130236592681, 0,
1.58126280198489359130236592681, 2.44409693399062889598354321935, 2.95219300692453178298894419696, 3.85266391878274018026939905146, 4.84564013692593615220452440900, 5.57042614644503521028691688059, 6.06287431340825476662839498481, 7.33099138824343281718675482683, 7.47194377080697428615717467585
