L(s) = 1 | + (1.35 + 1.07i)3-s + (1.70 − 1.44i)5-s − 5.00·7-s + (0.676 + 2.92i)9-s + 5.13·11-s + 5.63i·13-s + (3.87 − 0.121i)15-s + 1.95i·17-s + 3.00i·19-s + (−6.78 − 5.39i)21-s + (1.84 − 4.42i)23-s + (0.819 − 4.93i)25-s + (−2.23 + 4.69i)27-s + 4.77i·29-s − 2.36·31-s + ⋯ |
L(s) = 1 | + (0.782 + 0.622i)3-s + (0.762 − 0.646i)5-s − 1.89·7-s + (0.225 + 0.974i)9-s + 1.54·11-s + 1.56i·13-s + (0.999 − 0.0313i)15-s + 0.474i·17-s + 0.689i·19-s + (−1.48 − 1.17i)21-s + (0.385 − 0.922i)23-s + (0.163 − 0.986i)25-s + (−0.429 + 0.902i)27-s + 0.886i·29-s − 0.424·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.129054491\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.129054491\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 5 | \( 1 + (-1.70 + 1.44i)T \) |
| 23 | \( 1 + (-1.84 + 4.42i)T \) |
good | 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 5.63iT - 13T^{2} \) |
| 17 | \( 1 - 1.95iT - 17T^{2} \) |
| 19 | \( 1 - 3.00iT - 19T^{2} \) |
| 29 | \( 1 - 4.77iT - 29T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 2.97iT - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 5.69iT - 53T^{2} \) |
| 59 | \( 1 + 9.47iT - 59T^{2} \) |
| 61 | \( 1 - 4.89iT - 61T^{2} \) |
| 67 | \( 1 - 2.27T + 67T^{2} \) |
| 71 | \( 1 - 6.85iT - 71T^{2} \) |
| 73 | \( 1 - 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 4.83iT - 79T^{2} \) |
| 83 | \( 1 - 0.561iT - 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−9.588830095459308736413848481491, −8.994308946650982399890058035523, −8.701123124017318956861455498842, −7.07338441607164219500177778312, −6.50373796321389944719726088011, −5.68940359377554795133307335175, −4.28051487354573109125448851711, −3.85071705446000321791839659547, −2.68947845578116723263517327794, −1.51161487765241068748322595065,
0.827643791367987758044297102396, 2.40290266242560812736489912802, 3.19096122149900415862988995717, 3.77074428478598887611622613621, 5.65755405358735911135121403663, 6.28748757787942336886529173007, 6.96309021287090346887152374012, 7.54946568301985080539890404364, 8.984489126805878952227457507451, 9.327878313792254855410168715028