| L(s) = 1 | + 9.30e3·2-s + 7.70e5·3-s + 5.30e7·4-s + 2.44e8·5-s + 7.17e9·6-s + 2.22e10·7-s + 1.81e11·8-s − 2.53e11·9-s + 2.27e12·10-s + 1.44e13·11-s + 4.08e13·12-s − 3.60e13·13-s + 2.07e14·14-s + 1.88e14·15-s − 9.10e13·16-s − 3.16e14·17-s − 2.35e15·18-s − 1.22e16·19-s + 1.29e16·20-s + 1.71e16·21-s + 1.34e17·22-s + 2.08e17·23-s + 1.39e17·24-s + 5.96e16·25-s − 3.35e17·26-s − 8.48e17·27-s + 1.18e18·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.837·3-s + 1.58·4-s + 0.447·5-s + 1.34·6-s + 0.608·7-s + 0.933·8-s − 0.299·9-s + 0.718·10-s + 1.38·11-s + 1.32·12-s − 0.428·13-s + 0.977·14-s + 0.374·15-s − 0.0808·16-s − 0.131·17-s − 0.480·18-s − 1.27·19-s + 0.707·20-s + 0.509·21-s + 2.23·22-s + 1.98·23-s + 0.781·24-s + 0.199·25-s − 0.688·26-s − 1.08·27-s + 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(6.516070996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.516070996\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 2.44e8T \) |
| good | 2 | \( 1 - 9.30e3T + 3.35e7T^{2} \) |
| 3 | \( 1 - 7.70e5T + 8.47e11T^{2} \) |
| 7 | \( 1 - 2.22e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.44e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 3.60e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.16e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.22e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 2.08e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.80e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 5.74e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.65e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 3.56e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.39e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.08e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.50e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 8.29e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.43e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 1.61e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.38e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.91e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 3.10e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.19e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.58e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.37e24T + 4.66e49T^{2} \) |
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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
−17.12651538364768910732623895680, −14.84974346958895188906393238912, −14.37565018398247086277712419771, −12.99058166856029008629741642746, −11.38470570076156472252448621444, −8.883424148080948577689516878726, −6.61842423525362991846485652995, −4.88857269544941328827608428084, −3.37300503519884592595418562183, −1.94964119840963392273652671465,
1.94964119840963392273652671465, 3.37300503519884592595418562183, 4.88857269544941328827608428084, 6.61842423525362991846485652995, 8.883424148080948577689516878726, 11.38470570076156472252448621444, 12.99058166856029008629741642746, 14.37565018398247086277712419771, 14.84974346958895188906393238912, 17.12651538364768910732623895680
