| L(s) = 1 | + 2.82i·5-s − 4.47i·7-s − 3.16·11-s + 13-s + 4.24i·17-s − 6.32·23-s − 3.00·25-s + 4.24i·29-s + 8.94i·31-s + 12.6·35-s + 2·37-s + 5.65i·41-s − 3.16·47-s − 13.0·49-s + 1.41i·53-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s − 1.69i·7-s − 0.953·11-s + 0.277·13-s + 1.02i·17-s − 1.31·23-s − 0.600·25-s + 0.787i·29-s + 1.60i·31-s + 2.13·35-s + 0.328·37-s + 0.883i·41-s − 0.461·47-s − 1.85·49-s + 0.194i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8656557049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8656557049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8.94iT - 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−9.936651961988033128030713982370, −8.481666891003482499953707529791, −7.83851007643207570299332665499, −7.06113277618002610538512092200, −6.58955639936696525440695488995, −5.58820563798723813071711235157, −4.38398970681063026525777202800, −3.63571377473198697800959887505, −2.81488563558976188699598000071, −1.43450590244404833217221603212,
0.30996800475777155817225770121, 1.97581970291656523060122757789, 2.72056743274735132346944523314, 4.12807958517192733587481209025, 5.09388962172029747667625941558, 5.56991153247823128985983279338, 6.30033337066023907094524305789, 7.76206671171728769941322740395, 8.181486175587602737855594118704, 9.043864151284050173457565585994
