| L(s) = 1 | + 0.498·2-s − 0.240·3-s − 1.75·4-s − 0.581·5-s − 0.120·6-s − 1.41·7-s − 1.87·8-s − 2.94·9-s − 0.290·10-s + 11-s + 0.421·12-s − 0.703·14-s + 0.139·15-s + 2.56·16-s + 0.451·17-s − 1.46·18-s − 3.19·19-s + 1.01·20-s + 0.339·21-s + 0.498·22-s − 4.03·23-s + 0.450·24-s − 4.66·25-s + 1.42·27-s + 2.46·28-s + 10.2·29-s + 0.0698·30-s + ⋯ |
| L(s) = 1 | + 0.352·2-s − 0.138·3-s − 0.875·4-s − 0.260·5-s − 0.0490·6-s − 0.532·7-s − 0.661·8-s − 0.980·9-s − 0.0917·10-s + 0.301·11-s + 0.121·12-s − 0.188·14-s + 0.0361·15-s + 0.642·16-s + 0.109·17-s − 0.346·18-s − 0.731·19-s + 0.227·20-s + 0.0740·21-s + 0.106·22-s − 0.841·23-s + 0.0919·24-s − 0.932·25-s + 0.275·27-s + 0.466·28-s + 1.89·29-s + 0.0127·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9529887739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9529887739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.498T + 2T^{2} \) |
| 3 | \( 1 + 0.240T + 3T^{2} \) |
| 5 | \( 1 + 0.581T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 17 | \( 1 - 0.451T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 0.0779T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 2.84T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−9.232010648197240242380464180193, −8.378541315337200616307451065176, −7.976534469371372568353851707581, −6.50928420972748494548832672277, −6.09262330107880990837246925866, −5.11722398103918937287668313127, −4.27581700315644906212730109770, −3.48793332345944304098925263779, −2.51020549194201023627694971842, −0.61453258534102479974295723372,
0.61453258534102479974295723372, 2.51020549194201023627694971842, 3.48793332345944304098925263779, 4.27581700315644906212730109770, 5.11722398103918937287668313127, 6.09262330107880990837246925866, 6.50928420972748494548832672277, 7.976534469371372568353851707581, 8.378541315337200616307451065176, 9.232010648197240242380464180193
