| L(s) = 1 | + 4.92e3·2-s − 1.40e6·3-s − 9.31e6·4-s − 7.35e8·5-s − 6.93e9·6-s − 2.11e11·8-s + 1.13e12·9-s − 3.61e12·10-s − 1.13e13·11-s + 1.31e13·12-s − 1.01e14·13-s + 1.03e15·15-s − 7.26e14·16-s − 2.51e15·17-s + 5.59e15·18-s + 6.23e15·19-s + 6.84e15·20-s − 5.58e16·22-s + 1.46e17·23-s + 2.97e17·24-s + 2.42e17·25-s − 5.02e17·26-s − 4.07e17·27-s − 2.07e18·29-s + 5.09e18·30-s − 2.66e18·31-s + 3.50e18·32-s + ⋯ |
| L(s) = 1 | + 0.850·2-s − 1.53·3-s − 0.277·4-s − 1.34·5-s − 1.30·6-s − 1.08·8-s + 1.34·9-s − 1.14·10-s − 1.08·11-s + 0.424·12-s − 1.21·13-s + 2.06·15-s − 0.645·16-s − 1.04·17-s + 1.14·18-s + 0.646·19-s + 0.373·20-s − 0.925·22-s + 1.39·23-s + 1.66·24-s + 0.812·25-s − 1.03·26-s − 0.522·27-s − 1.08·29-s + 1.75·30-s − 0.607·31-s + 0.537·32-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 | \( 1 - 4.92e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.40e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 7.35e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 1.13e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.01e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.51e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 6.23e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.46e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.07e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 2.66e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 3.73e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 3.19e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 4.31e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.31e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.15e20T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.26e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.77e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 2.66e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 8.63e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.34e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 8.12e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.15e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 4.18e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.72e24T + 4.66e49T^{2} \) |
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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
−10.92623489835684430470316544293, −9.400791269367964516656122584377, −7.79832109514830920719144164361, −6.81040328870903613769004718858, −5.41619647969600989529025119094, −4.89986855827313568467780488206, −4.00935063996153409089639141603, −2.71429407748417672573776960656, −0.61020539656947531901146300501, 0,
0.61020539656947531901146300501, 2.71429407748417672573776960656, 4.00935063996153409089639141603, 4.89986855827313568467780488206, 5.41619647969600989529025119094, 6.81040328870903613769004718858, 7.79832109514830920719144164361, 9.400791269367964516656122584377, 10.92623489835684430470316544293
