L(s) = 1 | − 3.46i·7-s + 3.46i·11-s + (−1 − 3.46i)13-s + 6·17-s + 3.46i·19-s + 5·25-s − 6·29-s − 3.46i·31-s − 6.92i·37-s − 6.92i·41-s + 4·43-s − 3.46i·47-s − 4.99·49-s − 6·53-s − 10.3i·59-s + ⋯ |
L(s) = 1 | − 1.30i·7-s + 1.04i·11-s + (−0.277 − 0.960i)13-s + 1.45·17-s + 0.794i·19-s + 25-s − 1.11·29-s − 0.622i·31-s − 1.13i·37-s − 1.08i·41-s + 0.609·43-s − 0.505i·47-s − 0.714·49-s − 0.824·53-s − 1.35i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.591566889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.591566889\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 3.46i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−9.268123451807267301344687037471, −7.918041400392443925458204203521, −7.62082698432222449838988599889, −6.90737143399282830056499369999, −5.77437843594195012621508984265, −5.01262964210276623961677455470, −3.99395706974002866084525692113, −3.31003592284693946367836947711, −1.90102068319411022057398772856, −0.64240583080349214362919923775,
1.28076768259758631432022814307, 2.62737972260535610336860269209, 3.30140874401848639117646589066, 4.61039473510964536743426527653, 5.45263091749220973143811563412, 6.08989652046962761771009124632, 6.97813910636055363684576624627, 7.944543224238885824206823459200, 8.754121734649953833262045227067, 9.209366869335164030777677561765