L(s) = 1 | − 2·2-s + (−4.86 + 1.82i)3-s + 4·4-s + 6.14i·5-s + (9.72 − 3.65i)6-s − 15.5·7-s − 8·8-s + (20.3 − 17.7i)9-s − 12.2i·10-s + 37.5·11-s + (−19.4 + 7.31i)12-s + 29.6i·13-s + 31.0·14-s + (−11.2 − 29.8i)15-s + 16·16-s − 47.7i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.936 + 0.351i)3-s + 0.5·4-s + 0.549i·5-s + (0.661 − 0.248i)6-s − 0.837·7-s − 0.353·8-s + (0.752 − 0.658i)9-s − 0.388i·10-s + 1.03·11-s + (−0.468 + 0.175i)12-s + 0.632i·13-s + 0.592·14-s + (−0.193 − 0.514i)15-s + 0.250·16-s − 0.681i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6747561671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6747561671\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (4.86 - 1.82i)T \) |
| 59 | \( 1 + (-25.0 - 452. i)T \) |
good | 5 | \( 1 - 6.14iT - 125T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 - 37.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 47.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.79T + 1.21e4T^{2} \) |
| 29 | \( 1 + 129. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 12.3iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 237. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 336. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 347. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 173.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 99.3iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 764. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 596. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 755. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 360. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 963.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765. iT - 9.12e5T^{2} \) |
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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
−11.47494297971548896581403191062, −10.22369301164868092233976493226, −9.686446925963523068059114861728, −8.860376694189981279810848909767, −7.20083491169422993360815866294, −6.69105990372411132552573191484, −5.77184118193747769070410036307, −4.28665576689219484100080529446, −3.01355503971671039796516152714, −1.10376664937780530868071775478,
0.42044281230418058877634863789, 1.55481975663707001915529286580, 3.45643733571945333956145736709, 5.00566405347742426890997646590, 6.11189710843640241654990798471, 6.83587152175607332135226061386, 7.888868030917485508131300011739, 9.018031944237262965128929030287, 9.828618079646785852482759540068, 10.72850811485309832935002665542