| L(s) = 1 | − 11.3·2-s − 27·3-s + 1.07·4-s + 392.·5-s + 306.·6-s − 1.17e3·7-s + 1.44e3·8-s + 729·9-s − 4.46e3·10-s + 1.76e3·11-s − 29.0·12-s − 1.48e4·13-s + 1.33e4·14-s − 1.06e4·15-s − 1.65e4·16-s − 2.61e4·17-s − 8.28e3·18-s + 4.24e4·19-s + 423.·20-s + 3.16e4·21-s − 2.00e4·22-s − 1.21e4·23-s − 3.89e4·24-s + 7.60e4·25-s + 1.68e5·26-s − 1.96e4·27-s − 1.26e3·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.577·3-s + 0.00841·4-s + 1.40·5-s + 0.579·6-s − 1.29·7-s + 0.995·8-s + 0.333·9-s − 1.41·10-s + 0.399·11-s − 0.00485·12-s − 1.87·13-s + 1.29·14-s − 0.811·15-s − 1.00·16-s − 1.29·17-s − 0.334·18-s + 1.41·19-s + 0.0118·20-s + 0.744·21-s − 0.401·22-s − 0.208·23-s − 0.574·24-s + 0.974·25-s + 1.88·26-s − 0.192·27-s − 0.0108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7374982181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7374982181\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 11.3T + 128T^{2} \) |
| 5 | \( 1 - 392.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.17e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.76e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.48e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.24e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.66e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.06e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.88e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.41e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.24e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.16e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.86e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.87e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.63e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.48e7T + 8.07e13T^{2} \) |
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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
−13.28571304469701222247182214070, −12.18729519998584721056088845897, −10.50164969495625436678525725584, −9.647906392791937698925891288471, −9.284974944181860900928264965180, −7.29364350095003659401289746712, −6.22553270778339561513586546790, −4.79769354988430986928577338477, −2.39240665409515027701553841911, −0.68489013352477954903216523346,
0.68489013352477954903216523346, 2.39240665409515027701553841911, 4.79769354988430986928577338477, 6.22553270778339561513586546790, 7.29364350095003659401289746712, 9.284974944181860900928264965180, 9.647906392791937698925891288471, 10.50164969495625436678525725584, 12.18729519998584721056088845897, 13.28571304469701222247182214070
