L(s) = 1 | + i·2-s − 4-s + 1.63i·5-s + (1.91 + 1.82i)7-s − i·8-s − 1.63·10-s + 2.23·11-s + 0.801·13-s + (−1.82 + 1.91i)14-s + 16-s − 2.71i·17-s + (3.13 − 3.02i)19-s − 1.63i·20-s + 2.23i·22-s + 6.49·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.733i·5-s + (0.722 + 0.691i)7-s − 0.353i·8-s − 0.518·10-s + 0.672·11-s + 0.222·13-s + (−0.488 + 0.511i)14-s + 0.250·16-s − 0.658i·17-s + (0.719 − 0.694i)19-s − 0.366i·20-s + 0.475i·22-s + 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00530 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00530 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.109997521\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.109997521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
| 19 | \( 1 + (-3.13 + 3.02i)T \) |
good | 5 | \( 1 - 1.63iT - 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 0.801T + 13T^{2} \) |
| 17 | \( 1 + 2.71iT - 17T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + 7.19iT - 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 - 1.91iT - 47T^{2} \) |
| 53 | \( 1 + 1.19iT - 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 0.934iT - 61T^{2} \) |
| 67 | \( 1 + 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 1.19iT - 71T^{2} \) |
| 73 | \( 1 - 6.99iT - 73T^{2} \) |
| 79 | \( 1 + 0.503iT - 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−9.055582859161645133685296891669, −8.423682426849946384147080963699, −7.40764696292420551832484341618, −6.95111266546293318401971968959, −6.15075624104782457382715591456, −5.21848372753642705464850363279, −4.66026792543890537458120322305, −3.40556359080239774682170724632, −2.57241241564687306848898410373, −1.10995661286983100124435716463,
0.979511673635380387343205485207, 1.54304038692568576815094097915, 2.98947720362489912038544332361, 3.97238703433868905184088721797, 4.62551630159952798938039009526, 5.36973253335058707688805008737, 6.42321454377153593162876433332, 7.38268789343475280845423306339, 8.266770799149815901626303628803, 8.729055884267475836783195266865