L(s) = 1 | − 2-s + 4-s − 5-s + 5.13·7-s − 8-s + 10-s − 2.22·11-s − 6.60·13-s − 5.13·14-s + 16-s − 5.63·17-s − 7.51·19-s − 20-s + 2.22·22-s + 6.10·23-s + 25-s + 6.60·26-s + 5.13·28-s + 7.29·29-s − 0.967·31-s − 32-s + 5.63·34-s − 5.13·35-s − 37-s + 7.51·38-s + 40-s + 1.03·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.94·7-s − 0.353·8-s + 0.316·10-s − 0.669·11-s − 1.83·13-s − 1.37·14-s + 0.250·16-s − 1.36·17-s − 1.72·19-s − 0.223·20-s + 0.473·22-s + 1.27·23-s + 0.200·25-s + 1.29·26-s + 0.970·28-s + 1.35·29-s − 0.173·31-s − 0.176·32-s + 0.966·34-s − 0.867·35-s − 0.164·37-s + 1.21·38-s + 0.158·40-s + 0.161·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.151225470\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.151225470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 + 0.967T + 31T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 + 0.329T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 - 2.27T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.530817886082246053498332584175, −8.033319120547349293742334392427, −7.24766422220181163282554022056, −6.78903206791551795668425354360, −5.41305324396109122843952938560, −4.75211608781613119658309285907, −4.24082500344435270043478823358, −2.43910836535835245514511280788, −2.22118524646101579447834432215, −0.69168407326170216395107205197,
0.69168407326170216395107205197, 2.22118524646101579447834432215, 2.43910836535835245514511280788, 4.24082500344435270043478823358, 4.75211608781613119658309285907, 5.41305324396109122843952938560, 6.78903206791551795668425354360, 7.24766422220181163282554022056, 8.033319120547349293742334392427, 8.530817886082246053498332584175