| L(s) = 1 | − 662.·2-s − 1.07e6·3-s − 3.31e7·4-s − 3.68e8·5-s + 7.15e8·6-s + 4.41e10·8-s + 3.18e11·9-s + 2.44e11·10-s − 8.55e12·11-s + 3.57e13·12-s + 6.10e13·13-s + 3.98e14·15-s + 1.08e15·16-s − 1.62e15·17-s − 2.11e14·18-s + 5.88e15·19-s + 1.22e16·20-s + 5.67e15·22-s − 7.30e15·23-s − 4.76e16·24-s − 1.62e17·25-s − 4.04e16·26-s + 5.70e17·27-s − 1.90e18·29-s − 2.63e17·30-s + 5.51e18·31-s − 2.19e18·32-s + ⋯ |
| L(s) = 1 | − 0.114·2-s − 1.17·3-s − 0.986·4-s − 0.675·5-s + 0.134·6-s + 0.227·8-s + 0.376·9-s + 0.0772·10-s − 0.822·11-s + 1.15·12-s + 0.726·13-s + 0.792·15-s + 0.960·16-s − 0.675·17-s − 0.0430·18-s + 0.609·19-s + 0.666·20-s + 0.0940·22-s − 0.0694·23-s − 0.266·24-s − 0.544·25-s − 0.0830·26-s + 0.731·27-s − 0.997·29-s − 0.0906·30-s + 1.25·31-s − 0.337·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.06243701450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06243701450\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 662.T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.07e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 3.68e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 8.55e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 6.10e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.62e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 5.88e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 7.30e15T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.90e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 5.51e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.99e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.97e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 3.21e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 9.08e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 2.67e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 2.40e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 5.08e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.28e23T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.59e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.37e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 2.41e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.80e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.70e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 7.03e23T + 4.66e49T^{2} \) |
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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
−10.99779766869088451241507706478, −9.955700309040160775640787502592, −8.633689488231891248910840611273, −7.68737575472262375712270432422, −6.23560983303868858164143505175, −5.23384771089397900682080774570, −4.40919487627425726042163906883, −3.22632256702280174786690715067, −1.34974905724012633373561967006, −0.12527844413031739941923098733,
0.12527844413031739941923098733, 1.34974905724012633373561967006, 3.22632256702280174786690715067, 4.40919487627425726042163906883, 5.23384771089397900682080774570, 6.23560983303868858164143505175, 7.68737575472262375712270432422, 8.633689488231891248910840611273, 9.955700309040160775640787502592, 10.99779766869088451241507706478
