L(s) = 1 | − 2·2-s + (4.75 + 2.10i)3-s + 4·4-s + 16.7i·5-s + (−9.50 − 4.20i)6-s − 28.7·7-s − 8·8-s + (18.1 + 19.9i)9-s − 33.5i·10-s + 44.5·11-s + (19.0 + 8.40i)12-s + 14.6i·13-s + 57.5·14-s + (−35.2 + 79.7i)15-s + 16·16-s + 32.9i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.914 + 0.404i)3-s + 0.5·4-s + 1.50i·5-s + (−0.646 − 0.285i)6-s − 1.55·7-s − 0.353·8-s + (0.673 + 0.739i)9-s − 1.06i·10-s + 1.22·11-s + (0.457 + 0.202i)12-s + 0.313i·13-s + 1.09·14-s + (−0.607 + 1.37i)15-s + 0.250·16-s + 0.470i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.051149623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051149623\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-4.75 - 2.10i)T \) |
| 59 | \( 1 + (430. - 142. i)T \) |
good | 5 | \( 1 - 16.7iT - 125T^{2} \) |
| 7 | \( 1 + 28.7T + 343T^{2} \) |
| 11 | \( 1 - 44.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 32.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 38.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 61.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 372. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 44.8iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 152. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 202. iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 608. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 240. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 980. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 76.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 147.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 890.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10463090394967560470848969751, −10.29420702463571114094660350375, −9.652697206012628735568803429922, −8.970101589527584788042274227796, −7.73926587994612628597457867480, −6.71727312181279989861687544462, −6.29050976329925047987646229025, −3.87846844311577117390180513072, −3.22292780119749001000046159166, −2.07062827770015037011811971972,
0.40128596027252360844844646503, 1.59529128562207361872033062289, 3.14944509735980145573987889393, 4.28787608589725837098558312894, 6.05539187204969772191772206418, 6.86315414371444656350320183992, 8.080559111845531415734213085398, 8.828902036395636347340989215468, 9.484151285224503509914362201032, 10.00926629562477341013462347505