| L(s) = 1 | + 2.55·2-s − 2.41·3-s + 4.53·4-s − 5-s − 6.17·6-s + 3.41·7-s + 6.48·8-s + 2.84·9-s − 2.55·10-s − 5.46·11-s − 10.9·12-s + 5.38·13-s + 8.72·14-s + 2.41·15-s + 7.50·16-s − 2.83·17-s + 7.26·18-s − 1.20·19-s − 4.53·20-s − 8.24·21-s − 13.9·22-s + 7.33·23-s − 15.6·24-s + 25-s + 13.7·26-s + 0.381·27-s + 15.4·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.39·3-s + 2.26·4-s − 0.447·5-s − 2.52·6-s + 1.28·7-s + 2.29·8-s + 0.947·9-s − 0.808·10-s − 1.64·11-s − 3.16·12-s + 1.49·13-s + 2.33·14-s + 0.624·15-s + 1.87·16-s − 0.686·17-s + 1.71·18-s − 0.275·19-s − 1.01·20-s − 1.80·21-s − 2.97·22-s + 1.52·23-s − 3.20·24-s + 0.200·25-s + 2.69·26-s + 0.0733·27-s + 2.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.047028200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.047028200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 37 | \( 1 - 2.54T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 - 1.90T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 3.97T + 79T^{2} \) |
| 83 | \( 1 + 1.32T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 6.87T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.074227596127163390322684530390, −7.10417645341874198812613516838, −6.60944420091150976667185096014, −5.68107150759065572925115617625, −5.31132514666122411926611035477, −4.73675341331815652463334751242, −4.16522985980020430832600509384, −3.12090503759730527314482094258, −2.15562329756327105277797943027, −0.928780998297197598368446726862,
0.928780998297197598368446726862, 2.15562329756327105277797943027, 3.12090503759730527314482094258, 4.16522985980020430832600509384, 4.73675341331815652463334751242, 5.31132514666122411926611035477, 5.68107150759065572925115617625, 6.60944420091150976667185096014, 7.10417645341874198812613516838, 8.074227596127163390322684530390
