| L(s) = 1 | + 0.00817·2-s − 0.445·3-s − 1.99·4-s − 3.45·5-s − 0.00363·6-s − 1.06·7-s − 0.0327·8-s − 2.80·9-s − 0.0282·10-s + 1.20·11-s + 0.890·12-s + 1.53·13-s − 0.00868·14-s + 1.53·15-s + 3.99·16-s + 0.0987·17-s − 0.0229·18-s − 6.08·19-s + 6.91·20-s + 0.472·21-s + 0.00982·22-s + 0.357·23-s + 0.0145·24-s + 6.95·25-s + 0.0125·26-s + 2.58·27-s + 2.12·28-s + ⋯ |
| L(s) = 1 | + 0.00578·2-s − 0.256·3-s − 0.999·4-s − 1.54·5-s − 0.00148·6-s − 0.401·7-s − 0.0115·8-s − 0.933·9-s − 0.00893·10-s + 0.362·11-s + 0.256·12-s + 0.424·13-s − 0.00232·14-s + 0.397·15-s + 0.999·16-s + 0.0239·17-s − 0.00539·18-s − 1.39·19-s + 1.54·20-s + 0.103·21-s + 0.00209·22-s + 0.0744·23-s + 0.00297·24-s + 1.39·25-s + 0.00245·26-s + 0.496·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5102085452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5102085452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 787 | \( 1 - T \) |
| good | 2 | \( 1 - 0.00817T + 2T^{2} \) |
| 3 | \( 1 + 0.445T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.0987T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 - 0.357T + 23T^{2} \) |
| 29 | \( 1 - 5.47T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 4.07T + 59T^{2} \) |
| 61 | \( 1 - 0.772T + 61T^{2} \) |
| 67 | \( 1 - 0.401T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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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
−10.42405305271656331849023209183, −9.225746908431225107790453862210, −8.463169621906313974100657666389, −8.056125394889583429597633283451, −6.79109177865438629927512465520, −5.86097818084777284215557610368, −4.63157744308192670905716955033, −3.99143089999125962135307082543, −3.01737842059580651682638764259, −0.58081025361693766335701677893,
0.58081025361693766335701677893, 3.01737842059580651682638764259, 3.99143089999125962135307082543, 4.63157744308192670905716955033, 5.86097818084777284215557610368, 6.79109177865438629927512465520, 8.056125394889583429597633283451, 8.463169621906313974100657666389, 9.225746908431225107790453862210, 10.42405305271656331849023209183
