| L(s) = 1 | + 0.606·2-s + 0.930·3-s − 7.63·4-s − 5·5-s + 0.563·6-s − 1.50·7-s − 9.47·8-s − 26.1·9-s − 3.03·10-s + 49.3·11-s − 7.09·12-s − 13·13-s − 0.909·14-s − 4.65·15-s + 55.3·16-s − 111.·17-s − 15.8·18-s + 20.3·19-s + 38.1·20-s − 1.39·21-s + 29.9·22-s + 192.·23-s − 8.81·24-s + 25·25-s − 7.88·26-s − 49.4·27-s + 11.4·28-s + ⋯ |
| L(s) = 1 | + 0.214·2-s + 0.178·3-s − 0.954·4-s − 0.447·5-s + 0.0383·6-s − 0.0810·7-s − 0.418·8-s − 0.967·9-s − 0.0958·10-s + 1.35·11-s − 0.170·12-s − 0.277·13-s − 0.0173·14-s − 0.0800·15-s + 0.864·16-s − 1.59·17-s − 0.207·18-s + 0.245·19-s + 0.426·20-s − 0.0145·21-s + 0.290·22-s + 1.74·23-s − 0.0749·24-s + 0.200·25-s − 0.0594·26-s − 0.352·27-s + 0.0773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 0.606T + 8T^{2} \) |
| 3 | \( 1 - 0.930T + 27T^{2} \) |
| 7 | \( 1 + 1.50T + 343T^{2} \) |
| 11 | \( 1 - 49.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.0T + 2.43e4T^{2} \) |
| 37 | \( 1 - 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 450.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 440.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 199.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 180.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 431.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 843.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 953.T + 9.12e5T^{2} \) |
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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
−8.604920620700711854679212927057, −7.74735128488970416911427811548, −6.74758672480230204302422474388, −6.02432900695631595336134440096, −4.97379915958526717476718017663, −4.32286442677165231041020004452, −3.50078169596378919520106841486, −2.60543881539060314667323858172, −1.05491189394376976124007910411, 0,
1.05491189394376976124007910411, 2.60543881539060314667323858172, 3.50078169596378919520106841486, 4.32286442677165231041020004452, 4.97379915958526717476718017663, 6.02432900695631595336134440096, 6.74758672480230204302422474388, 7.74735128488970416911427811548, 8.604920620700711854679212927057
