| L(s) = 1 | + (−0.309 − 0.951i)2-s + (2.59 − 1.88i)3-s + (−0.809 + 0.587i)4-s + (−0.0648 − 2.23i)5-s + (−2.59 − 1.88i)6-s + 7-s + (0.809 + 0.587i)8-s + (2.25 − 6.93i)9-s + (−2.10 + 0.752i)10-s + (1.17 + 3.61i)11-s + (−0.991 + 3.05i)12-s + (−1.68 + 5.19i)13-s + (−0.309 − 0.951i)14-s + (−4.38 − 5.67i)15-s + (0.309 − 0.951i)16-s + (−4.75 − 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.49 − 1.08i)3-s + (−0.404 + 0.293i)4-s + (−0.0290 − 0.999i)5-s + (−1.05 − 0.769i)6-s + 0.377·7-s + (0.286 + 0.207i)8-s + (0.751 − 2.31i)9-s + (−0.665 + 0.237i)10-s + (0.353 + 1.08i)11-s + (−0.286 + 0.880i)12-s + (−0.467 + 1.44i)13-s + (−0.0825 − 0.254i)14-s + (−1.13 − 1.46i)15-s + (0.0772 − 0.237i)16-s + (−1.15 − 0.838i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945068 - 1.60680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945068 - 1.60680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.0648 + 2.23i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (-2.59 + 1.88i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 3.61i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.68 - 5.19i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.75 + 3.45i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.35 - 3.16i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.697 - 2.14i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.79 - 2.75i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.13 - 3.00i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.131 + 0.403i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.25 + 6.94i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + (0.214 - 0.155i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.20 + 3.78i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.61 + 8.05i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.57 - 4.84i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.86 + 7.16i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (6.35 - 4.61i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.54 - 7.83i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.00 + 3.63i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.35 - 4.61i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.135 - 0.415i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.44 - 2.49i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.70763856785256982190941591961, −9.756510956519315093906089706419, −9.216256231014783503242499841404, −8.603578701666338627670254413141, −7.53402097218056876521723111310, −6.89887388710654294646050136710, −4.83597711736525823848262779733, −3.79085636468112921047059844316, −2.21157317336748541989769452712, −1.48977121541007538470471186829,
2.61587443992294497052645453710, 3.52967230830175752564534802181, 4.69774449956730408865257029898, 6.01094356017390503433603536836, 7.41083729468156755142943770550, 8.149617429842308529848986288802, 8.860909661094900073241122514330, 9.852675579007510608821084989579, 10.54449912123751417524790776228, 11.34383941863628666312167099818
