| L(s) = 1 | + 2.06·2-s − 2.54·3-s + 2.27·4-s + 0.855·5-s − 5.26·6-s + 0.572·8-s + 3.47·9-s + 1.76·10-s − 6.04·11-s − 5.79·12-s − 2.17·15-s − 3.36·16-s + 4.70·17-s + 7.18·18-s + 2.96·19-s + 1.94·20-s − 12.5·22-s + 3.25·23-s − 1.45·24-s − 4.26·25-s − 1.20·27-s + 4.50·29-s − 4.50·30-s + 1.94·31-s − 8.11·32-s + 15.3·33-s + 9.72·34-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 1.46·3-s + 1.13·4-s + 0.382·5-s − 2.14·6-s + 0.202·8-s + 1.15·9-s + 0.559·10-s − 1.82·11-s − 1.67·12-s − 0.562·15-s − 0.842·16-s + 1.14·17-s + 1.69·18-s + 0.681·19-s + 0.435·20-s − 2.66·22-s + 0.678·23-s − 0.297·24-s − 0.853·25-s − 0.231·27-s + 0.835·29-s − 0.822·30-s + 0.348·31-s − 1.43·32-s + 2.67·33-s + 1.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066967849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066967849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.06T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 - 0.855T + 5T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 + 7.96T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 9.74T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 - 3.57T + 73T^{2} \) |
| 79 | \( 1 + 0.811T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.52043675851494696301846947900, −6.71055787196381234600657543902, −6.13079970716449708083835522704, −5.47414149511916940934330661527, −5.08022835061613131471096261624, −4.79803790004418910013618868853, −3.53677455994094635476433343996, −2.96856777236552190969174391773, −1.93733000449971820282617026511, −0.58905122835830236514679685709,
0.58905122835830236514679685709, 1.93733000449971820282617026511, 2.96856777236552190969174391773, 3.53677455994094635476433343996, 4.79803790004418910013618868853, 5.08022835061613131471096261624, 5.47414149511916940934330661527, 6.13079970716449708083835522704, 6.71055787196381234600657543902, 7.52043675851494696301846947900
