| L(s) = 1 | + 2.01i·5-s − 2.64·7-s − 2.01i·11-s − 7.34i·17-s − 3.35·19-s + 9.36i·23-s + 0.937·25-s + 2.29·31-s − 5.33i·35-s + 11.9·37-s − 9.36i·41-s + 7.00·49-s + 4.06·55-s − 16.7i·71-s + 5.33i·77-s + ⋯ |
| L(s) = 1 | + 0.901i·5-s − 0.999·7-s − 0.607i·11-s − 1.78i·17-s − 0.769·19-s + 1.95i·23-s + 0.187·25-s + 0.411·31-s − 0.901i·35-s + 1.96·37-s − 1.46i·41-s + 49-s + 0.547·55-s − 1.98i·71-s + 0.607i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.423413890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.423413890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 2.01iT - 5T^{2} \) |
| 11 | \( 1 + 2.01iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 9.36iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 9.36iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 16.7iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.942361838760175885343664376163, −7.73290384044356327482037142158, −7.22953840896219303200986513283, −6.46054152224062639764582792566, −5.85580522876473977515431457918, −4.91200924686703184812761832225, −3.73244079010820176763374269038, −3.09350204979369570807572078654, −2.35202285888336777694489587132, −0.64939869264194115234745103004,
0.78213556610527361847047356801, 2.05333297566573759586525239512, 3.05475326010469232405578264598, 4.36131327197173892145004326711, 4.47479126498541948284134646127, 5.91693641559906708360944033214, 6.29519061502422504097346805851, 7.13942039037987873125192010254, 8.383866320236161089818891693052, 8.454987433426359864865703047937
