L(s) = 1 | − 3.91i·5-s + (2.03 − 1.68i)7-s + 1.62·11-s − 2.54·13-s − 2.94·17-s − 2.72·19-s + 8.57i·23-s − 10.3·25-s + 8.21·29-s − 8.08i·31-s + (−6.61 − 7.98i)35-s − 8.05i·37-s − 9.17·41-s − 9.05i·43-s − 1.17·47-s + ⋯ |
L(s) = 1 | − 1.75i·5-s + (0.769 − 0.638i)7-s + 0.490·11-s − 0.706·13-s − 0.713·17-s − 0.624·19-s + 1.78i·23-s − 2.07·25-s + 1.52·29-s − 1.45i·31-s + (−1.11 − 1.34i)35-s − 1.32i·37-s − 1.43·41-s − 1.38i·43-s − 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.177425084\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.177425084\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.03 + 1.68i)T \) |
good | 5 | \( 1 + 3.91iT - 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 + 8.08iT - 31T^{2} \) |
| 37 | \( 1 + 8.05iT - 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 + 9.05iT - 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 - 1.45iT - 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 7.35iT - 67T^{2} \) |
| 71 | \( 1 + 7.71iT - 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 + 0.0913T + 79T^{2} \) |
| 83 | \( 1 + 2.03iT - 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 + 7.53iT - 97T^{2} \) |
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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
−8.083362986129043973690109270623, −7.56664303410807472251839152863, −6.65152185847334996615470422760, −5.60975303496257496091444535120, −4.98708874082267812180777618578, −4.36448681531075384385000979058, −3.75849987785556668632965492182, −2.13605063533807903956394126294, −1.38584069291683787946654495241, −0.32417897692329822359565749074,
1.68117856826391069655164292545, 2.65902862614212411080778432795, 3.05831206467557614497771701637, 4.40804126887618269091491150609, 4.89920145666267736643172782026, 6.22494908626311754463846915412, 6.56797096238390713511176605885, 7.13684643953431855975485228263, 8.230896367333445341359683785020, 8.549444622421261023472570071838