| L(s) = 1 | + 5.49e3·2-s + 1.52e6·3-s − 3.39e6·4-s − 1.00e9·5-s + 8.39e9·6-s − 2.02e11·8-s + 1.48e12·9-s − 5.54e12·10-s + 1.39e13·11-s − 5.17e12·12-s − 6.33e13·13-s − 1.54e15·15-s − 1.00e15·16-s − 6.78e14·17-s + 8.16e15·18-s − 1.52e16·19-s + 3.42e15·20-s + 7.65e16·22-s + 7.85e16·23-s − 3.10e17·24-s + 7.21e17·25-s − 3.48e17·26-s + 9.77e17·27-s − 1.15e18·29-s − 8.47e18·30-s − 8.59e17·31-s + 1.31e18·32-s + ⋯ |
| L(s) = 1 | + 0.948·2-s + 1.65·3-s − 0.101·4-s − 1.84·5-s + 1.57·6-s − 1.04·8-s + 1.75·9-s − 1.75·10-s + 1.33·11-s − 0.167·12-s − 0.754·13-s − 3.07·15-s − 0.888·16-s − 0.282·17-s + 1.66·18-s − 1.58·19-s + 0.186·20-s + 1.26·22-s + 0.747·23-s − 1.73·24-s + 2.42·25-s − 0.715·26-s + 1.25·27-s − 0.604·29-s − 2.91·30-s − 0.195·31-s + 0.201·32-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
| \(L(13)\) |
\(\approx\) |
\(3.761337746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.761337746\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 - 5.49e3T + 3.35e7T^{2} \) |
| 3 | \( 1 - 1.52e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 1.00e9T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.39e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 6.33e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 6.78e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.52e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 7.85e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.15e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 8.59e17T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.89e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.33e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.57e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 5.03e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 5.46e20T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.15e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 4.67e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 8.62e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.69e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.82e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.19e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 9.97e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.70e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 2.07e24T + 4.66e49T^{2} \) |
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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
−11.24461108739868401128315855420, −9.329950898404699218488980365080, −8.657478751335281883220891094526, −7.72260272815968577979134627409, −6.66216130811821832024427624212, −4.54270769515684463628212051640, −4.04533242261230304299270517882, −3.36453894626365357336947356016, −2.34152193395742032414979315323, −0.63727493308412796638631847023,
0.63727493308412796638631847023, 2.34152193395742032414979315323, 3.36453894626365357336947356016, 4.04533242261230304299270517882, 4.54270769515684463628212051640, 6.66216130811821832024427624212, 7.72260272815968577979134627409, 8.657478751335281883220891094526, 9.329950898404699218488980365080, 11.24461108739868401128315855420
