| L(s) = 1 | + 2-s − 2.48·3-s + 4-s − 2.96·5-s − 2.48·6-s + 2.86·7-s + 8-s + 3.15·9-s − 2.96·10-s − 11-s − 2.48·12-s + 1.15·13-s + 2.86·14-s + 7.35·15-s + 16-s + 7.28·17-s + 3.15·18-s − 2.96·20-s − 7.11·21-s − 22-s + 5.73·23-s − 2.48·24-s + 3.77·25-s + 1.15·26-s − 0.387·27-s + 2.86·28-s − 1.38·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.43·3-s + 0.5·4-s − 1.32·5-s − 1.01·6-s + 1.08·7-s + 0.353·8-s + 1.05·9-s − 0.936·10-s − 0.301·11-s − 0.716·12-s + 0.320·13-s + 0.766·14-s + 1.89·15-s + 0.250·16-s + 1.76·17-s + 0.743·18-s − 0.662·20-s − 1.55·21-s − 0.213·22-s + 1.19·23-s − 0.506·24-s + 0.755·25-s + 0.226·26-s − 0.0746·27-s + 0.542·28-s − 0.257·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743727162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743727162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 7.28T + 17T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 + 0.906T + 47T^{2} \) |
| 53 | \( 1 + 4.96T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 - 2.66T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.70690244161773950010040824663, −7.15977042112763143458948083588, −6.27547135666425119799987197612, −5.56832804308581033084264941512, −5.03193868782788365118838474115, −4.53519395118940788334129010470, −3.71750958390914318509906962988, −2.97156895743206457809874320326, −1.50090630076029300836934688342, −0.68700383959030718329347893577,
0.68700383959030718329347893577, 1.50090630076029300836934688342, 2.97156895743206457809874320326, 3.71750958390914318509906962988, 4.53519395118940788334129010470, 5.03193868782788365118838474115, 5.56832804308581033084264941512, 6.27547135666425119799987197612, 7.15977042112763143458948083588, 7.70690244161773950010040824663
