L(s) = 1 | + (−0.880 + 0.880i)3-s + (−1.04 + 1.04i)7-s + 1.44i·9-s − 2.44·11-s + (3.03 − 3.03i)13-s + (4.66 − 4.66i)17-s + (4.11 − 1.44i)19-s − 1.84i·21-s + (−3.61 − 3.61i)23-s + (−3.91 − 3.91i)27-s + 8.22·29-s + 10.0i·31-s + (2.15 − 2.15i)33-s + (5.19 + 5.19i)37-s + 5.34i·39-s + ⋯ |
L(s) = 1 | + (−0.508 + 0.508i)3-s + (−0.396 + 0.396i)7-s + 0.483i·9-s − 0.738·11-s + (0.842 − 0.842i)13-s + (1.13 − 1.13i)17-s + (0.943 − 0.332i)19-s − 0.403i·21-s + (−0.754 − 0.754i)23-s + (−0.753 − 0.753i)27-s + 1.52·29-s + 1.80i·31-s + (0.375 − 0.375i)33-s + (0.853 + 0.853i)37-s + 0.856i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350571964\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350571964\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.11 + 1.44i)T \) |
good | 3 | \( 1 + (0.880 - 0.880i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.04 - 1.04i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + (-3.03 + 3.03i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.66 + 4.66i)T - 17iT^{2} \) |
| 23 | \( 1 + (3.61 + 3.61i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.84iT - 41T^{2} \) |
| 43 | \( 1 + (5.71 + 5.71i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.04 + 1.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.95 - 6.95i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.34T + 61T^{2} \) |
| 67 | \( 1 + (-6.55 - 6.55i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.84iT - 71T^{2} \) |
| 73 | \( 1 + (-9.33 - 9.33i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 + (-12.4 - 12.4i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.22T + 89T^{2} \) |
| 97 | \( 1 + (10.4 + 10.4i)T + 97iT^{2} \) |
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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
−9.534197675489684488850793664938, −8.405754624808518207117320947961, −7.930842186965978444971285416824, −6.90406293081519625164358456027, −5.93081686315723411164302990597, −5.25106314712987582361897503952, −4.69949400778360110536495991872, −3.29435605008392189361129868211, −2.68018439114234688620247852689, −0.919395067247455861177000634146,
0.73493304091693700922122158273, 1.83808298059408003824410825166, 3.36379118490034268976034461021, 3.95717625248241323636469190598, 5.25531481745683180267439269924, 6.15083545016714231822030563815, 6.45853606489715544857528319078, 7.68470906111410501350950808778, 8.005881649286419057928947177527, 9.247122449012032412376796445338