L(s) = 1 | + (−0.887 − 1.10i)2-s + (−0.707 − 0.707i)3-s + (−0.425 + 1.95i)4-s − 3.47·5-s + (−0.151 + 1.40i)6-s + (3.04 + 3.04i)7-s + (2.52 − 1.26i)8-s + 1.00i·9-s + (3.08 + 3.82i)10-s − 2.48·11-s + (1.68 − 1.08i)12-s + (−2.36 − 2.72i)13-s + (0.651 − 6.05i)14-s + (2.45 + 2.45i)15-s + (−3.63 − 1.66i)16-s − 0.873i·17-s + ⋯ |
L(s) = 1 | + (−0.627 − 0.778i)2-s + (−0.408 − 0.408i)3-s + (−0.212 + 0.977i)4-s − 1.55·5-s + (−0.0617 + 0.574i)6-s + (1.15 + 1.15i)7-s + (0.894 − 0.447i)8-s + 0.333i·9-s + (0.974 + 1.20i)10-s − 0.749·11-s + (0.485 − 0.312i)12-s + (−0.655 − 0.755i)13-s + (0.174 − 1.61i)14-s + (0.634 + 0.634i)15-s + (−0.909 − 0.415i)16-s − 0.211i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.246595 - 0.466195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.246595 - 0.466195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.887 + 1.10i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (2.36 + 2.72i)T \) |
good | 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 + (-3.04 - 3.04i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 17 | \( 1 + 0.873iT - 17T^{2} \) |
| 19 | \( 1 - 8.32T + 19T^{2} \) |
| 23 | \( 1 + 2.16iT - 23T^{2} \) |
| 29 | \( 1 + (-2.02 + 2.02i)T - 29iT^{2} \) |
| 31 | \( 1 + (7.24 + 7.24i)T + 31iT^{2} \) |
| 37 | \( 1 + 4.91iT - 37T^{2} \) |
| 41 | \( 1 + (-4.08 - 4.08i)T + 41iT^{2} \) |
| 43 | \( 1 + (7.47 + 7.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.88 + 2.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.46 + 2.46i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.67T + 59T^{2} \) |
| 61 | \( 1 + (-1.16 + 1.16i)T - 61iT^{2} \) |
| 67 | \( 1 - 5.85T + 67T^{2} \) |
| 71 | \( 1 + (1.42 - 1.42i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.72 + 1.72i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.43iT - 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + (-10.4 + 10.4i)T - 89iT^{2} \) |
| 97 | \( 1 + (0.456 - 0.456i)T - 97iT^{2} \) |
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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
−10.59721213046908643452088295496, −9.414166668798989161578304840066, −8.387411979096223735693084690515, −7.71931260765807148287863282814, −7.41659295007004451060234998589, −5.45890452795709691854515305711, −4.71187807852485406212226389057, −3.32344824535158868005497160470, −2.21202786960111987339980093594, −0.45637318751531335795124329113,
1.15099754343077146384588443341, 3.62280329072854172802805339753, 4.74211335972188783596764758212, 5.14183431766645255695860820597, 6.85735963095935902143904390652, 7.59528510301477667524820007940, 7.893699612588472911749107525433, 9.053690161105739479860782620197, 10.08965911615503715011705272454, 10.93354393483640312564405483042