L(s) = 1 | + 2.05i·3-s − i·5-s − 0.327·7-s − 1.24·9-s − 0.575·11-s + 0.243·13-s + 2.05·15-s + 0.756i·17-s − 5.67·19-s − 0.675i·21-s + (−4.36 + 1.98i)23-s − 25-s + 3.61i·27-s + 3.18·29-s + 10.3i·31-s + ⋯ |
L(s) = 1 | + 1.18i·3-s − 0.447i·5-s − 0.123·7-s − 0.414·9-s − 0.173·11-s + 0.0674·13-s + 0.531·15-s + 0.183i·17-s − 1.30·19-s − 0.147i·21-s + (−0.910 + 0.412i)23-s − 0.200·25-s + 0.696i·27-s + 0.592·29-s + 1.86i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7905264770\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7905264770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (4.36 - 1.98i)T \) |
good | 3 | \( 1 - 2.05iT - 3T^{2} \) |
| 7 | \( 1 + 0.327T + 7T^{2} \) |
| 11 | \( 1 + 0.575T + 11T^{2} \) |
| 13 | \( 1 - 0.243T + 13T^{2} \) |
| 17 | \( 1 - 0.756iT - 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 6.24iT - 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 0.655T + 43T^{2} \) |
| 47 | \( 1 - 6.09iT - 47T^{2} \) |
| 53 | \( 1 - 5.75iT - 53T^{2} \) |
| 59 | \( 1 - 0.921iT - 59T^{2} \) |
| 61 | \( 1 - 12.1iT - 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 8.06iT - 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 2.48iT - 89T^{2} \) |
| 97 | \( 1 - 1.65iT - 97T^{2} \) |
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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
−9.708969751354232675066353154439, −8.834147424427862215928968992844, −8.401503629260043615222912471477, −7.29015933533632183064741113464, −6.31733683564854060355398688012, −5.42261855334783501449358775704, −4.61265673817055104628930109297, −3.99552038108231562244451350443, −3.03811961361973834713564483135, −1.63991727657831619604645567217,
0.27616763972032133614618164779, 1.79769347529210448725323096573, 2.52749719102951446220598875854, 3.75282985725256426003561311032, 4.77414610808703371908606882144, 6.04615160804009015186392969937, 6.51016268325660802122400577637, 7.19991468961854339174537041279, 8.116326770438833448464751597859, 8.487633155721139316383835118812