| L(s) = 1 | − 2-s − 7·4-s + 5i·5-s + 14i·7-s + 15·8-s − 5i·10-s + 20i·11-s − 58·13-s − 14i·14-s + 41·16-s + (17 + 68i)17-s + 80·19-s − 35i·20-s − 20i·22-s + 118i·23-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.875·4-s + 0.447i·5-s + 0.755i·7-s + 0.662·8-s − 0.158i·10-s + 0.548i·11-s − 1.23·13-s − 0.267i·14-s + 0.640·16-s + (0.242 + 0.970i)17-s + 0.965·19-s − 0.391i·20-s − 0.193i·22-s + 1.06i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5366311546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5366311546\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| 17 | \( 1 + (-17 - 68i)T \) |
| good | 2 | \( 1 + T + 8T^{2} \) |
| 7 | \( 1 - 14iT - 343T^{2} \) |
| 11 | \( 1 - 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 58T + 2.19e3T^{2} \) |
| 19 | \( 1 - 80T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 126iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 70iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 134iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 100iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 272T + 7.95e4T^{2} \) |
| 47 | \( 1 - 464T + 1.03e5T^{2} \) |
| 53 | \( 1 + 642T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180T + 2.05e5T^{2} \) |
| 61 | \( 1 + 110iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 924T + 3.00e5T^{2} \) |
| 71 | \( 1 - 90iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 828iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.33e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 552T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3iT - 9.12e5T^{2} \) |
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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
−10.10264783119262187619488115721, −9.571977700571254408161654879961, −8.844084924294353468627204017788, −7.78557468793819287209879099751, −7.22448373063385657719412863422, −5.81531775741286154918724597497, −5.10003192353599843728172474124, −4.02294700994622702641276324468, −2.82455133886230251918919275996, −1.50241701518432985227705302432,
0.20962525888679202069555978487, 1.02728935031978317871780645521, 2.78905387016857489907687681912, 4.12399991486893525580221325055, 4.83218900968117299092467381835, 5.73251330401217675200044233329, 7.21321938599776353510892862110, 7.73611160198586805057705096925, 8.739723734303078658696717873776, 9.495609738217990034205910716316
