L(s) = 1 | + 3i·3-s + (2 + i)5-s + i·7-s − 6·9-s − 3·11-s + i·13-s + (−3 + 6i)15-s − 5i·17-s − 8·19-s − 3·21-s − 2i·23-s + (3 + 4i)25-s − 9i·27-s − 29-s − 2·31-s + ⋯ |
L(s) = 1 | + 1.73i·3-s + (0.894 + 0.447i)5-s + 0.377i·7-s − 2·9-s − 0.904·11-s + 0.277i·13-s + (−0.774 + 1.54i)15-s − 1.21i·17-s − 1.83·19-s − 0.654·21-s − 0.417i·23-s + (0.600 + 0.800i)25-s − 1.73i·27-s − 0.185·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6699323260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6699323260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 11iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−9.492987320535211944225704902250, −9.111238874354766857775725947845, −8.344252088362326939530888561360, −7.17049084866963700037214922295, −6.15412376120825316987270052478, −5.47360379222520665015900734106, −4.80577473880434330016221863908, −3.99292374455575562478518197491, −2.87101040072881044861217000711, −2.27269886395836717496666217794,
0.20538396839752688489011916643, 1.60717004862425648696223033430, 2.07269256668331964944122702958, 3.23289674997376481972549278331, 4.68037806565601783885755275343, 5.61785934618745828041184879912, 6.33901539545124092961496511289, 6.78227434288184604338158613137, 7.921697050757963596148628509978, 8.235037415385111809874040158856