| L(s) = 1 | + 2-s − 2.64·3-s + 4-s + 3.64·5-s − 2.64·6-s − 1.64·7-s + 8-s + 4.00·9-s + 3.64·10-s + 0.645·11-s − 2.64·12-s + 2·13-s − 1.64·14-s − 9.64·15-s + 16-s + 4.00·18-s + 3.64·20-s + 4.35·21-s + 0.645·22-s + 3.64·23-s − 2.64·24-s + 8.29·25-s + 2·26-s − 2.64·27-s − 1.64·28-s − 3.64·29-s − 9.64·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.63·5-s − 1.08·6-s − 0.622·7-s + 0.353·8-s + 1.33·9-s + 1.15·10-s + 0.194·11-s − 0.763·12-s + 0.554·13-s − 0.439·14-s − 2.49·15-s + 0.250·16-s + 0.942·18-s + 0.815·20-s + 0.950·21-s + 0.137·22-s + 0.760·23-s − 0.540·24-s + 1.65·25-s + 0.392·26-s − 0.509·27-s − 0.311·28-s − 0.676·29-s − 1.76·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860496710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860496710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.645T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 + 0.354T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.93T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−10.64211105744574206196313988813, −9.790167171953430464729858219660, −9.025203936843283371721843901917, −7.26146694989215338827916550489, −6.38492634696085628431120923884, −5.89607768974149006289580234569, −5.34318984845976856398573077045, −4.21870548115609127283573686009, −2.67352286044052877188370057742, −1.22689914664726047037100613067,
1.22689914664726047037100613067, 2.67352286044052877188370057742, 4.21870548115609127283573686009, 5.34318984845976856398573077045, 5.89607768974149006289580234569, 6.38492634696085628431120923884, 7.26146694989215338827916550489, 9.025203936843283371721843901917, 9.790167171953430464729858219660, 10.64211105744574206196313988813
