L(s) = 1 | + (0.281 − 0.959i)3-s + (0.415 + 0.909i)5-s + (0.909 − 0.415i)7-s + (−0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.281 + 0.959i)13-s + (0.989 − 0.142i)15-s + (0.142 − 0.989i)17-s + (0.540 − 0.841i)19-s + (−0.142 − 0.989i)21-s + (0.540 − 0.841i)23-s + (−0.654 + 0.755i)25-s + (−0.755 + 0.654i)27-s + (0.909 − 0.415i)29-s + (0.540 + 0.841i)31-s + ⋯ |
L(s) = 1 | + (0.281 − 0.959i)3-s + (0.415 + 0.909i)5-s + (0.909 − 0.415i)7-s + (−0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.281 + 0.959i)13-s + (0.989 − 0.142i)15-s + (0.142 − 0.989i)17-s + (0.540 − 0.841i)19-s + (−0.142 − 0.989i)21-s + (0.540 − 0.841i)23-s + (−0.654 + 0.755i)25-s + (−0.755 + 0.654i)27-s + (0.909 − 0.415i)29-s + (0.540 + 0.841i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.784354822 - 0.4377184186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784354822 - 0.4377184186i\) |
\(L(1)\) |
\(\approx\) |
\(1.324629539 - 0.2288938309i\) |
\(L(1)\) |
\(\approx\) |
\(1.324629539 - 0.2288938309i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.909 - 0.415i)T \) |
| 11 | \( 1 + (-0.415 + 0.909i)T \) |
| 13 | \( 1 + (-0.281 + 0.959i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (0.540 - 0.841i)T \) |
| 23 | \( 1 + (0.540 - 0.841i)T \) |
| 29 | \( 1 + (0.909 - 0.415i)T \) |
| 31 | \( 1 + (0.540 + 0.841i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.281 + 0.959i)T \) |
| 43 | \( 1 + (0.909 + 0.415i)T \) |
| 47 | \( 1 + (0.959 - 0.281i)T \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.281 + 0.959i)T \) |
| 61 | \( 1 + (0.755 - 0.654i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.989 - 0.142i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.35224439566668972676982883891, −21.68681140180993525714198293784, −20.94744487614588792523183338797, −20.588517153558469474847340549, −19.59142410480056588590919126983, −18.64587529220910483799329550334, −17.33877859978051165250395588709, −17.15162731646875581353131195497, −15.91308327412187477375135171289, −15.46399494327968373345057309150, −14.41876720763583231726508465991, −13.75032328941886727555958648880, −12.73343480988708833276594416407, −11.788710670059333477996279003682, −10.80508162460902416965162911972, −10.08929062262629779000413113955, −9.127882964939189226774150658, −8.28917979689671653996898120875, −7.897352035301567672507805628818, −5.80577052594197602662867218930, −5.47145696841503840767036066886, −4.5683518021789519504971178599, −3.48453296521848614679640506590, −2.41561291175721872281911914303, −1.12102558544202672872161023275,
1.09575232552059193295918715588, 2.28406881250685995087697265460, 2.78605267384566331040506163598, 4.36386212375222739724444241046, 5.31511684469269536776286417557, 6.695698365813735933002084237245, 7.09286020560965245155138687153, 7.86037147083015037778003390633, 9.016234022486758215342460880686, 9.92016697554175932883466776395, 11.023014306242334866781599020132, 11.66674968676235184516194446324, 12.6142039057470732690397186383, 13.69737793979469660455617296012, 14.20030551753406322927524838869, 14.77567323114295591492247686818, 15.901616929370659119724021160387, 17.26165617433047669708985628333, 17.76619208971904276343947759338, 18.36579730647140871933633087254, 19.16534640331302905395944352364, 20.06950142818537169177602430425, 20.86660612739628620473681151590, 21.62934385322616692355384079734, 22.87477887892980206256859004504