| L(s) = 1 | − 3-s − 3.37·5-s + 0.792i·7-s + 9-s + 0.792i·11-s + 5.04i·13-s + 3.37·15-s + 5.37·17-s + (−4 − 1.73i)19-s − 0.792i·21-s − 8.51i·23-s + 6.37·25-s − 27-s − 10.0i·29-s − 8.74·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.50·5-s + 0.299i·7-s + 0.333·9-s + 0.238i·11-s + 1.40i·13-s + 0.870·15-s + 1.30·17-s + (−0.917 − 0.397i)19-s − 0.172i·21-s − 1.77i·23-s + 1.27·25-s − 0.192·27-s − 1.87i·29-s − 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.424336 - 0.378157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424336 - 0.378157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 - 0.792iT - 7T^{2} \) |
| 11 | \( 1 - 0.792iT - 11T^{2} \) |
| 13 | \( 1 - 5.04iT - 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 5.04iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 9.30iT - 43T^{2} \) |
| 47 | \( 1 + 4.25iT - 47T^{2} \) |
| 53 | \( 1 - 3.16iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 13.2iT - 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−10.03433111311526134307336233548, −8.914589279638326545119700472105, −8.224312407863223103464847712898, −7.25546198935187917360656634540, −6.63740818499600890420239376467, −5.47645914520473065222480776230, −4.33359954585422204844628530313, −3.87052911292979782953202490879, −2.24759979608745054011809640632, −0.35143735854871599649762726020,
1.11186385128181932382564624403, 3.30120663720140239022297223840, 3.81026836997235737790763411114, 5.10485937446315768105789629432, 5.79677153032710855568665066029, 7.14839097121117454128937685430, 7.69244088283222367184622452531, 8.341903421944930656544637237173, 9.548343033430614883443187772432, 10.55485100624293765874571116242
