L(s) = 1 | + (0.103 − 1.41i)2-s + (1.29 + 1.15i)3-s + (−1.97 − 0.291i)4-s + (0.186 + 0.186i)5-s + (1.75 − 1.70i)6-s + 7-s + (−0.615 + 2.76i)8-s + (0.350 + 2.97i)9-s + (0.282 − 0.243i)10-s + (4.29 − 4.29i)11-s + (−2.22 − 2.65i)12-s + (2.73 + 2.73i)13-s + (0.103 − 1.41i)14-s + (0.0267 + 0.455i)15-s + (3.83 + 1.15i)16-s − 4.98i·17-s + ⋯ |
L(s) = 1 | + (0.0730 − 0.997i)2-s + (0.747 + 0.664i)3-s + (−0.989 − 0.145i)4-s + (0.0833 + 0.0833i)5-s + (0.717 − 0.696i)6-s + 0.377·7-s + (−0.217 + 0.976i)8-s + (0.116 + 0.993i)9-s + (0.0892 − 0.0770i)10-s + (1.29 − 1.29i)11-s + (−0.642 − 0.766i)12-s + (0.759 + 0.759i)13-s + (0.0275 − 0.376i)14-s + (0.00691 + 0.117i)15-s + (0.957 + 0.288i)16-s − 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64188 - 0.531736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64188 - 0.531736i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.103 + 1.41i)T \) |
| 3 | \( 1 + (-1.29 - 1.15i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-0.186 - 0.186i)T + 5iT^{2} \) |
| 11 | \( 1 + (-4.29 + 4.29i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.98iT - 17T^{2} \) |
| 19 | \( 1 + (3.09 - 3.09i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.55iT - 23T^{2} \) |
| 29 | \( 1 + (-3.75 + 3.75i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.58iT - 31T^{2} \) |
| 37 | \( 1 + (4.82 - 4.82i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + (5.79 + 5.79i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.22T + 47T^{2} \) |
| 53 | \( 1 + (7.95 + 7.95i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.18 - 3.18i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.17 + 3.17i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.39 + 2.39i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.06iT - 71T^{2} \) |
| 73 | \( 1 - 2.32iT - 73T^{2} \) |
| 79 | \( 1 - 6.89iT - 79T^{2} \) |
| 83 | \( 1 + (1.12 + 1.12i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−11.46217450825242355175848554457, −10.54670460446893313961679771413, −9.677115303037132273840535151983, −8.700500799403295386664696840538, −8.368044195323701684925265226482, −6.51994340841124663146057668228, −5.10060242848866081372216975044, −3.98907957434536545688207220965, −3.19996173587952695080612078421, −1.64406122717306217379341184467,
1.56973864700296261387978634870, 3.57977057914585049628053575997, 4.62508926731623769059689839288, 6.16648018130995322295757936412, 6.84038863681352663407379511989, 7.83833841241859704963992940868, 8.672875810484226280222113634295, 9.320775929348096798812496260734, 10.51865024314347814439564154460, 12.05725308286520949064580245475