L(s) = 1 | − 5-s + 3.11i·7-s + 4.47·11-s + 3.14·13-s + 0.883·17-s + 5.10i·19-s + (2.92 + 3.80i)23-s + 25-s + 6.05i·29-s + 4.03·31-s − 3.11i·35-s + 1.93i·37-s + 7.43i·41-s + 7.84i·43-s − 10.5i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.17i·7-s + 1.35·11-s + 0.873·13-s + 0.214·17-s + 1.17i·19-s + (0.609 + 0.792i)23-s + 0.200·25-s + 1.12i·29-s + 0.724·31-s − 0.526i·35-s + 0.317i·37-s + 1.16i·41-s + 1.19i·43-s − 1.53i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0403 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.176909973\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.176909973\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-2.92 - 3.80i)T \) |
good | 7 | \( 1 - 3.11iT - 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 0.883T + 17T^{2} \) |
| 19 | \( 1 - 5.10iT - 19T^{2} \) |
| 29 | \( 1 - 6.05iT - 29T^{2} \) |
| 31 | \( 1 - 4.03T + 31T^{2} \) |
| 37 | \( 1 - 1.93iT - 37T^{2} \) |
| 41 | \( 1 - 7.43iT - 41T^{2} \) |
| 43 | \( 1 - 7.84iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 0.0758T + 53T^{2} \) |
| 59 | \( 1 + 6.25iT - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 + 16.2iT - 67T^{2} \) |
| 71 | \( 1 + 12.8iT - 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 + 3.66iT - 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 7.27iT - 97T^{2} \) |
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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
−8.099051734239449069438983921660, −7.32003400609517258956790005716, −6.39474813546071101323089575008, −6.09060353475913863636709466836, −5.22870911101391394343115069341, −4.46363726300280630918797297083, −3.50616310929317653308052898259, −3.15328130315819420704510056351, −1.82504505127986661918555559410, −1.16625969770455781288028098421,
0.61604535577140949120534999988, 1.20142808825903445148771451647, 2.50725247583490222220825438680, 3.48244837937178002958702299259, 4.16904269634792045198140104133, 4.47313315215659523309277268094, 5.62522959546014385196206124821, 6.44086151999351379599414451283, 6.97950811627186415507444464819, 7.45310597005428984653318354216