L(s) = 1 | − 2-s − 0.806·3-s + 4-s − 1.86·5-s + 0.806·6-s − 7-s − 8-s − 2.35·9-s + 1.86·10-s + 5.76·11-s − 0.806·12-s + 2.28·13-s + 14-s + 1.50·15-s + 16-s + 5.11·17-s + 2.35·18-s − 1.86·20-s + 0.806·21-s − 5.76·22-s + 5.31·23-s + 0.806·24-s − 1.50·25-s − 2.28·26-s + 4.31·27-s − 28-s + 6.54·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.465·3-s + 0.5·4-s − 0.835·5-s + 0.329·6-s − 0.377·7-s − 0.353·8-s − 0.783·9-s + 0.591·10-s + 1.73·11-s − 0.232·12-s + 0.634·13-s + 0.267·14-s + 0.388·15-s + 0.250·16-s + 1.24·17-s + 0.553·18-s − 0.417·20-s + 0.175·21-s − 1.22·22-s + 1.10·23-s + 0.164·24-s − 0.301·25-s − 0.448·26-s + 0.829·27-s − 0.188·28-s + 1.21·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9616665138\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9616665138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 2.54T + 31T^{2} \) |
| 37 | \( 1 + 0.962T + 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 + 8.54T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.46T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 8.51T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 9.53T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 1.86T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 6.54T + 97T^{2} \) |
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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
−8.390253374370513855961832985069, −7.53451897780389879261819102487, −6.80100388473421597440770533610, −6.24185442382774203275049371627, −5.53732386250706034140224679471, −4.47434039942118475447896131165, −3.52174615998200404658639161118, −3.05917712757446268228082413345, −1.49241220150498657485349514753, −0.65798919418505052297648981426,
0.65798919418505052297648981426, 1.49241220150498657485349514753, 3.05917712757446268228082413345, 3.52174615998200404658639161118, 4.47434039942118475447896131165, 5.53732386250706034140224679471, 6.24185442382774203275049371627, 6.80100388473421597440770533610, 7.53451897780389879261819102487, 8.390253374370513855961832985069