| L(s) = 1 | + 1.07e4·2-s − 1.64e6·3-s + 8.26e7·4-s + 5.33e8·5-s − 1.77e10·6-s + 5.29e11·8-s + 1.85e12·9-s + 5.75e12·10-s − 2.57e12·11-s − 1.35e14·12-s − 5.77e13·13-s − 8.77e14·15-s + 2.93e15·16-s − 1.56e15·17-s + 1.99e16·18-s − 6.12e15·19-s + 4.41e16·20-s − 2.77e16·22-s − 5.06e16·23-s − 8.70e17·24-s − 1.29e16·25-s − 6.23e17·26-s − 1.65e18·27-s + 5.40e17·29-s − 9.45e18·30-s − 5.98e18·31-s + 1.38e19·32-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 1.78·3-s + 2.46·4-s + 0.977·5-s − 3.32·6-s + 2.72·8-s + 2.18·9-s + 1.82·10-s − 0.247·11-s − 4.39·12-s − 0.688·13-s − 1.74·15-s + 2.60·16-s − 0.650·17-s + 4.06·18-s − 0.635·19-s + 2.40·20-s − 0.459·22-s − 0.481·23-s − 4.86·24-s − 0.0435·25-s − 1.28·26-s − 2.11·27-s + 0.283·29-s − 3.24·30-s − 1.36·31-s + 2.12·32-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
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 - 1.07e4T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.64e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 5.33e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 2.57e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 5.77e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.56e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 6.12e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 5.06e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 5.40e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 5.98e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.79e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 3.81e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.63e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 5.17e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 3.50e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.82e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.40e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 4.58e21T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.62e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.99e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 8.29e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.45e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 5.46e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.09e24T + 4.66e49T^{2} \) |
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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
−10.88656278417625842309345659502, −9.924291269057224154415378003912, −7.26305492075621217879440869587, −6.30539793078153045757244639312, −5.75475514891107392806389780449, −4.93365703038653794790701324497, −4.13541850941356574201616248293, −2.45681016110674281327404836390, −1.54562733469944124337771713873, 0,
1.54562733469944124337771713873, 2.45681016110674281327404836390, 4.13541850941356574201616248293, 4.93365703038653794790701324497, 5.75475514891107392806389780449, 6.30539793078153045757244639312, 7.26305492075621217879440869587, 9.924291269057224154415378003912, 10.88656278417625842309345659502
