L(s) = 1 | + (−2 − 2i)2-s + (12.4 − 12.4i)3-s + 8i·4-s + (−16.9 − 18.3i)5-s − 49.8·6-s + (−23.6 − 23.6i)7-s + (16 − 16i)8-s − 229. i·9-s + (−2.70 + 70.6i)10-s + 192.·11-s + (99.6 + 99.6i)12-s + (194. − 194. i)13-s + 94.5i·14-s + (−440. − 16.8i)15-s − 64·16-s + (−114. − 114. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (1.38 − 1.38i)3-s + 0.5i·4-s + (−0.679 − 0.733i)5-s − 1.38·6-s + (−0.482 − 0.482i)7-s + (0.250 − 0.250i)8-s − 2.83i·9-s + (−0.0270 + 0.706i)10-s + 1.58·11-s + (0.692 + 0.692i)12-s + (1.15 − 1.15i)13-s + 0.482i·14-s + (−1.95 − 0.0749i)15-s − 0.250·16-s + (−0.396 − 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.082313103\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082313103\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 2i)T \) |
| 5 | \( 1 + (16.9 + 18.3i)T \) |
| 23 | \( 1 + (77.9 - 77.9i)T \) |
good | 3 | \( 1 + (-12.4 + 12.4i)T - 81iT^{2} \) |
| 7 | \( 1 + (23.6 + 23.6i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 - 192.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-194. + 194. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (114. + 114. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 116. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 940. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 729.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.53e3 - 1.53e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.21e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (647. - 647. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-2.86e3 - 2.86e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (362. - 362. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 4.11e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.88e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-284. - 284. i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 2.86e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.30e3 + 2.30e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 1.64e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-474. + 474. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 2.41e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-8.04e3 - 8.04e3i)T + 8.85e7iT^{2} \) |
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
−11.27314430463142718312101846864, −9.619599185644966219278050070634, −8.841089339380895247368584709409, −8.226097761955976375471475282368, −7.30955717331558624510649685972, −6.37599874777780823713181301944, −3.86494524910419817192740029602, −3.22802120193567591240920084864, −1.46418008834825861975624386319, −0.76693394686434456795945124661,
2.20188158873418233395607497213, 3.71632649752420093645751264913, 4.20005139256724401148425475247, 6.13716351647017906453573303847, 7.26134373753412180740555013805, 8.577908303795700742099154759617, 8.973042350419457459539748157646, 9.793917654577299349976266689252, 10.86480016299886651041830088882, 11.63339748532401747794787517768