| L(s) = 1 | + 3.13i·3-s + i·5-s − 1.01·7-s − 6.80·9-s + 1.65·11-s − 4.63·13-s − 3.13·15-s − 6.97i·17-s − 4.61·19-s − 3.17i·21-s + (4.40 − 1.90i)23-s − 25-s − 11.9i·27-s + 0.837·29-s − 4.14i·31-s + ⋯ |
| L(s) = 1 | + 1.80i·3-s + 0.447i·5-s − 0.383·7-s − 2.26·9-s + 0.498·11-s − 1.28·13-s − 0.808·15-s − 1.69i·17-s − 1.05·19-s − 0.692i·21-s + (0.917 − 0.397i)23-s − 0.200·25-s − 2.29i·27-s + 0.155·29-s − 0.744i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1412648118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1412648118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.40 + 1.90i)T \) |
| good | 3 | \( 1 - 3.13iT - 3T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 6.97iT - 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 29 | \( 1 - 0.837T + 29T^{2} \) |
| 31 | \( 1 + 4.14iT - 31T^{2} \) |
| 37 | \( 1 - 3.47iT - 37T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 - 8.42T + 43T^{2} \) |
| 47 | \( 1 + 0.172iT - 47T^{2} \) |
| 53 | \( 1 + 2.29iT - 53T^{2} \) |
| 59 | \( 1 + 1.89iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.841iT - 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 8.11iT - 89T^{2} \) |
| 97 | \( 1 + 14.7iT - 97T^{2} \) |
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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
−9.200134775196414589013796947643, −8.762814555860406846478174490265, −7.49207556450720058623173068764, −6.67872975959523003321776097633, −5.67525857871442675460405032460, −4.76539028124409975769817432663, −4.31380413668408033734106495976, −3.14829700156515057421985906158, −2.59173486515129513719360629223, −0.05088216216608765163455857965,
1.35714849609577293147326344496, 2.14602550408010133234142048586, 3.25352820504869428854165113997, 4.54838834197416027947283379696, 5.68411958615718062233452381621, 6.36406778348583493474849095569, 7.02903850857618087461434335525, 7.71300028710221439257078381398, 8.529844561301799172537119521859, 9.047245704667694797770404951197
