| L(s) = 1 | − 296.·2-s − 2.18e3·3-s + 5.53e4·4-s + 6.49e5·6-s − 4.91e5·7-s − 6.68e6·8-s + 4.78e6·9-s + 9.58e6·11-s − 1.20e8·12-s − 1.59e8·13-s + 1.45e8·14-s + 1.72e8·16-s − 2.24e8·17-s − 1.41e9·18-s + 4.77e9·19-s + 1.07e9·21-s − 2.84e9·22-s + 3.94e8·23-s + 1.46e10·24-s + 4.72e10·26-s − 1.04e10·27-s − 2.71e10·28-s + 3.60e10·29-s + 2.24e11·31-s + 1.67e11·32-s − 2.09e10·33-s + 6.67e10·34-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.68·4-s + 0.946·6-s − 0.225·7-s − 1.12·8-s + 0.333·9-s + 0.148·11-s − 0.974·12-s − 0.703·13-s + 0.370·14-s + 0.160·16-s − 0.132·17-s − 0.546·18-s + 1.22·19-s + 0.130·21-s − 0.243·22-s + 0.0241·23-s + 0.650·24-s + 1.15·26-s − 0.192·27-s − 0.380·28-s + 0.388·29-s + 1.46·31-s + 0.864·32-s − 0.0856·33-s + 0.217·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.5739774513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5739774513\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2.18e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 296.T + 3.27e4T^{2} \) |
| 7 | \( 1 + 4.91e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 9.58e6T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.59e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.24e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 4.77e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 3.94e8T + 2.66e20T^{2} \) |
| 29 | \( 1 - 3.60e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.24e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.50e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 8.51e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 3.43e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.35e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.02e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 7.42e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.50e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.83e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 5.60e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.54e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.32e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.92e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.50e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 8.03e14T + 6.33e29T^{2} \) |
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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
−11.30697211264486140545059716568, −10.10088934007800469447000692081, −9.553300609144021625161418543897, −8.262830936112712721397414900436, −7.26783348409609337624333050207, −6.28645933861042584083454080321, −4.75937764926484267827603783539, −2.86126207272109736658551374563, −1.47460792070580892604693332798, −0.50699342626634997694737794435,
0.50699342626634997694737794435, 1.47460792070580892604693332798, 2.86126207272109736658551374563, 4.75937764926484267827603783539, 6.28645933861042584083454080321, 7.26783348409609337624333050207, 8.262830936112712721397414900436, 9.553300609144021625161418543897, 10.10088934007800469447000692081, 11.30697211264486140545059716568
