| L(s) = 1 | + (1.81 − 1.31i)5-s − 3.31i·7-s + 1.65·11-s + 5.05i·13-s + 7.05i·17-s − 1.24·19-s − 9.40i·23-s + (1.55 − 4.75i)25-s + 7.76·29-s + 2.25·31-s + (−4.34 − 5.99i)35-s − i·37-s + 3.25·41-s − 4.56i·43-s + 5.09i·47-s + ⋯ |
| L(s) = 1 | + (0.809 − 0.586i)5-s − 1.25i·7-s + 0.498·11-s + 1.40i·13-s + 1.71i·17-s − 0.286·19-s − 1.96i·23-s + (0.311 − 0.950i)25-s + 1.44·29-s + 0.405·31-s + (−0.734 − 1.01i)35-s − 0.164i·37-s + 0.507·41-s − 0.696i·43-s + 0.742i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.528423689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528423689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.81 + 1.31i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.31iT - 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 5.05iT - 13T^{2} \) |
| 17 | \( 1 - 7.05iT - 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 9.40iT - 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 41 | \( 1 - 3.25T + 41T^{2} \) |
| 43 | \( 1 + 4.56iT - 43T^{2} \) |
| 47 | \( 1 - 5.09iT - 47T^{2} \) |
| 53 | \( 1 + 4.36iT - 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + 0.459iT - 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 5.85iT - 73T^{2} \) |
| 79 | \( 1 + 5.88T + 79T^{2} \) |
| 83 | \( 1 - 2.42iT - 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.45iT - 97T^{2} \) |
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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
−8.028197069247099271812632399207, −6.96679110496147188682122319037, −6.42322494623360474419517098880, −6.07021642727814842169250004714, −4.68480437768455567591808556337, −4.43110015936705926401529621990, −3.71324053892477461124705131929, −2.39439740413560931477895157280, −1.60321236183409343457501249910, −0.74342051517531623394842866916,
0.983832725618381907132663379254, 2.14258692677088417984075791568, 2.89313771432514645742120469809, 3.33682386902213973064792158520, 4.79093431671622238661349977206, 5.35445191705251198308093558676, 5.94646260511378252270515764632, 6.55676720727722684504777401420, 7.42516663776808522689410550870, 8.016304205016517371310421567493
