| L(s) = 1 | + 0.661·2-s − 2.32·3-s − 1.56·4-s + 5-s − 1.53·6-s − 0.917·7-s − 2.35·8-s + 2.39·9-s + 0.661·10-s − 11-s + 3.62·12-s − 1.28·13-s − 0.607·14-s − 2.32·15-s + 1.56·16-s + 1.41·17-s + 1.58·18-s − 0.964·19-s − 1.56·20-s + 2.13·21-s − 0.661·22-s + 6.85·23-s + 5.47·24-s + 25-s − 0.849·26-s + 1.40·27-s + 1.43·28-s + ⋯ |
| L(s) = 1 | + 0.467·2-s − 1.34·3-s − 0.781·4-s + 0.447·5-s − 0.627·6-s − 0.346·7-s − 0.833·8-s + 0.797·9-s + 0.209·10-s − 0.301·11-s + 1.04·12-s − 0.355·13-s − 0.162·14-s − 0.599·15-s + 0.390·16-s + 0.343·17-s + 0.373·18-s − 0.221·19-s − 0.349·20-s + 0.464·21-s − 0.141·22-s + 1.43·23-s + 1.11·24-s + 0.200·25-s − 0.166·26-s + 0.271·27-s + 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 0.661T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 + 0.917T + 7T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 0.964T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.451T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 9.63T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 7.11T + 59T^{2} \) |
| 61 | \( 1 - 4.53T + 61T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.102002738040881013266730677880, −6.99939083240419777159397404378, −6.46027788758452482808285116688, −5.51388492052730032976890035148, −5.31238158320051001558676222519, −4.55118303947577651635047739318, −3.58798204590539788460123627796, −2.62159062484226992020040175088, −1.07589348576122014762534854734, 0,
1.07589348576122014762534854734, 2.62159062484226992020040175088, 3.58798204590539788460123627796, 4.55118303947577651635047739318, 5.31238158320051001558676222519, 5.51388492052730032976890035148, 6.46027788758452482808285116688, 6.99939083240419777159397404378, 8.102002738040881013266730677880
