| L(s) = 1 | + 0.775·2-s + 2.46·3-s − 1.39·4-s + 3.48·5-s + 1.91·6-s + 0.0624·7-s − 2.63·8-s + 3.09·9-s + 2.70·10-s + 3.55·11-s − 3.45·12-s − 0.927·13-s + 0.0484·14-s + 8.60·15-s + 0.756·16-s − 3.32·17-s + 2.39·18-s − 5.02·19-s − 4.87·20-s + 0.154·21-s + 2.75·22-s − 23-s − 6.50·24-s + 7.15·25-s − 0.718·26-s + 0.226·27-s − 0.0873·28-s + ⋯ |
| L(s) = 1 | + 0.548·2-s + 1.42·3-s − 0.699·4-s + 1.55·5-s + 0.780·6-s + 0.0236·7-s − 0.931·8-s + 1.03·9-s + 0.854·10-s + 1.07·11-s − 0.996·12-s − 0.257·13-s + 0.0129·14-s + 2.22·15-s + 0.189·16-s − 0.807·17-s + 0.564·18-s − 1.15·19-s − 1.09·20-s + 0.0336·21-s + 0.587·22-s − 0.208·23-s − 1.32·24-s + 1.43·25-s − 0.140·26-s + 0.0435·27-s − 0.0165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.196933581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.196933581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.775T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 - 0.0624T + 7T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 13 | \( 1 + 0.927T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.30T + 43T^{2} \) |
| 47 | \( 1 + 5.34T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| 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
−9.961315426392808802512317408534, −9.643014445333891165691728599792, −8.765186674834613460303784125941, −8.376774748109765174809904474573, −6.75827748060478943104023210791, −6.04909385634161460425972690084, −4.83218985754974822964648978207, −3.91440676206932076648916742083, −2.77171177054327030967030352099, −1.79591863825321990167706243192,
1.79591863825321990167706243192, 2.77171177054327030967030352099, 3.91440676206932076648916742083, 4.83218985754974822964648978207, 6.04909385634161460425972690084, 6.75827748060478943104023210791, 8.376774748109765174809904474573, 8.765186674834613460303784125941, 9.643014445333891165691728599792, 9.961315426392808802512317408534
