| L(s) = 1 | + 1.25·2-s + 1.78·3-s − 0.414·4-s + 5-s + 2.24·6-s − 0.828·7-s − 3.04·8-s + 0.171·9-s + 1.25·10-s + 2·11-s − 0.737·12-s + 4.29·13-s − 1.04·14-s + 1.78·15-s − 3·16-s + 7.65·17-s + 0.216·18-s − 0.414·20-s − 1.47·21-s + 2.51·22-s − 0.828·23-s − 5.41·24-s + 25-s + 5.41·26-s − 5.03·27-s + 0.343·28-s + 8.59·29-s + ⋯ |
| L(s) = 1 | + 0.890·2-s + 1.02·3-s − 0.207·4-s + 0.447·5-s + 0.915·6-s − 0.313·7-s − 1.07·8-s + 0.0571·9-s + 0.398·10-s + 0.603·11-s − 0.212·12-s + 1.19·13-s − 0.278·14-s + 0.459·15-s − 0.750·16-s + 1.85·17-s + 0.0509·18-s − 0.0926·20-s − 0.321·21-s + 0.536·22-s − 0.172·23-s − 1.10·24-s + 0.200·25-s + 1.06·26-s − 0.969·27-s + 0.0648·28-s + 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.760837977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.760837977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 3 | \( 1 - 1.78T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 - 0.737T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 6.81T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−9.311261933989550705112117303709, −8.436520714630898826840724303035, −7.986356977192196497805234799319, −6.58899569083391258859789035322, −6.00256137811455663746926578109, −5.18529548245026991869809452313, −4.06693877657075806410018931849, −3.36398341508515669174902282783, −2.76780591179750605253961810722, −1.23339406749050078765882619428,
1.23339406749050078765882619428, 2.76780591179750605253961810722, 3.36398341508515669174902282783, 4.06693877657075806410018931849, 5.18529548245026991869809452313, 6.00256137811455663746926578109, 6.58899569083391258859789035322, 7.986356977192196497805234799319, 8.436520714630898826840724303035, 9.311261933989550705112117303709
