| L(s) = 1 | + 12.4·2-s + 26.4·4-s − 463.·5-s − 63.1·7-s − 1.26e3·8-s − 5.75e3·10-s + 8.03e3·11-s + 1.52e3·13-s − 784.·14-s − 1.90e4·16-s + 2.86e4·17-s + 3.94e3·19-s − 1.22e4·20-s + 9.97e4·22-s + 8.27e4·23-s + 1.36e5·25-s + 1.90e4·26-s − 1.66e3·28-s − 1.94e5·29-s − 3.24e5·31-s − 7.53e4·32-s + 3.56e5·34-s + 2.92e4·35-s + 2.01e5·37-s + 4.89e4·38-s + 5.84e5·40-s − 859.·41-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.206·4-s − 1.65·5-s − 0.0695·7-s − 0.871·8-s − 1.82·10-s + 1.81·11-s + 0.193·13-s − 0.0763·14-s − 1.16·16-s + 1.41·17-s + 0.131·19-s − 0.342·20-s + 1.99·22-s + 1.41·23-s + 1.74·25-s + 0.212·26-s − 0.0143·28-s − 1.48·29-s − 1.95·31-s − 0.406·32-s + 1.55·34-s + 0.115·35-s + 0.654·37-s + 0.144·38-s + 1.44·40-s − 0.00194·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 - 12.4T + 128T^{2} \) |
| 5 | \( 1 + 463.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 63.1T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.52e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.94e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.94e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 859.T + 1.94e11T^{2} \) |
| 47 | \( 1 + 8.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.27e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.09e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.84e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.43e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.29e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.14e7T + 8.07e13T^{2} \) |
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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
−9.483225497421863427047283638134, −8.797298491798570568589474170344, −7.62115422711421488878967059402, −6.80568410500688130943422380388, −5.63838050567016721588872885829, −4.56956981952769361425404236892, −3.61793917300719094429585545487, −3.37831032265231326359934952750, −1.25146268427237081693713519388, 0,
1.25146268427237081693713519388, 3.37831032265231326359934952750, 3.61793917300719094429585545487, 4.56956981952769361425404236892, 5.63838050567016721588872885829, 6.80568410500688130943422380388, 7.62115422711421488878967059402, 8.797298491798570568589474170344, 9.483225497421863427047283638134
