| L(s) = 1 | + (−1.26 − 0.642i)2-s + (0.185 + 0.0294i)3-s + (1.17 + 1.61i)4-s + (4.66 − 1.78i)5-s + (−0.215 − 0.156i)6-s + (7.34 − 7.34i)7-s + (−0.442 − 2.79i)8-s + (−8.52 − 2.77i)9-s + (−7.03 − 0.745i)10-s + (5.47 + 16.8i)11-s + (0.170 + 0.335i)12-s + (7.48 − 3.81i)13-s + (−13.9 + 4.53i)14-s + (0.919 − 0.194i)15-s + (−1.23 + 3.80i)16-s + (−29.4 + 4.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (0.0619 + 0.00980i)3-s + (0.293 + 0.404i)4-s + (0.933 − 0.357i)5-s + (−0.0358 − 0.0260i)6-s + (1.04 − 1.04i)7-s + (−0.0553 − 0.349i)8-s + (−0.947 − 0.307i)9-s + (−0.703 − 0.0745i)10-s + (0.497 + 1.53i)11-s + (0.0142 + 0.0279i)12-s + (0.575 − 0.293i)13-s + (−0.997 + 0.324i)14-s + (0.0613 − 0.0129i)15-s + (−0.0772 + 0.237i)16-s + (−1.73 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.962746 - 0.308204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962746 - 0.308204i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.642i)T \) |
| 5 | \( 1 + (-4.66 + 1.78i)T \) |
| good | 3 | \( 1 + (-0.185 - 0.0294i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-7.34 + 7.34i)T - 49iT^{2} \) |
| 11 | \( 1 + (-5.47 - 16.8i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-7.48 + 3.81i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (29.4 - 4.66i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (9.36 - 12.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (3.14 - 6.17i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (13.3 + 18.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-13.0 - 9.49i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-0.504 - 0.991i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (14.4 - 44.5i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-8.20 - 8.20i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.42 - 34.2i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (31.2 + 4.94i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-58.9 - 19.1i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (19.9 + 61.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (59.8 - 9.47i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-66.2 + 48.1i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-23.7 + 46.6i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-1.37 - 1.88i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (15.3 + 96.8i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (6.69 - 2.17i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (1.05 - 6.63i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−15.11316731600636825336805838872, −14.05065332965705803275868514206, −12.91193526499473770288148138730, −11.47741725940908896917024223843, −10.43596438465923174567038574834, −9.241150225978815578101761171396, −8.069122637956690779811632193474, −6.47080268465897249385728005333, −4.45353424039347814005113981638, −1.79509629060796734913336638619,
2.32062181821523018607636021205, 5.43676744394116600913056904679, 6.47483268513067390497204147896, 8.659579301926652454019246430118, 8.855915693437316199853336095561, 10.95299490712534070447769827200, 11.40938180009772184374562985924, 13.55480698153396241757134031367, 14.33621927591682998720178613530, 15.42671578244935805892946223228
