| L(s) = 1 | + 1.21i·3-s − i·5-s + 4.59·7-s + 1.53·9-s − 3.99·11-s − 0.942·13-s + 1.21·15-s + 4.91i·17-s + 8.05·19-s + 5.56i·21-s + (1.82 − 4.43i)23-s − 25-s + 5.48i·27-s − 8.30·29-s + 5.27i·31-s + ⋯ |
| L(s) = 1 | + 0.698i·3-s − 0.447i·5-s + 1.73·7-s + 0.511·9-s − 1.20·11-s − 0.261·13-s + 0.312·15-s + 1.19i·17-s + 1.84·19-s + 1.21i·21-s + (0.379 − 0.925i)23-s − 0.200·25-s + 1.05i·27-s − 1.54·29-s + 0.947i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200204867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200204867\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.82 + 4.43i)T \) |
| good | 3 | \( 1 - 1.21iT - 3T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 0.942T + 13T^{2} \) |
| 17 | \( 1 - 4.91iT - 17T^{2} \) |
| 19 | \( 1 - 8.05T + 19T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 - 5.27iT - 31T^{2} \) |
| 37 | \( 1 + 3.06iT - 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 + 12.4iT - 47T^{2} \) |
| 53 | \( 1 - 3.64iT - 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 6.02T + 67T^{2} \) |
| 71 | \( 1 + 8.00iT - 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 + 1.22T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 - 4.19iT - 89T^{2} \) |
| 97 | \( 1 - 3.27iT - 97T^{2} \) |
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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
−9.233834473268490071618853945243, −8.628702347367011895774220725093, −7.59978226590432646350176600050, −7.46143177286351127778188804546, −5.71674952543045735873479076176, −5.15018187579681944974916478193, −4.56079603764589034707941726520, −3.66820260227686889219112791259, −2.26338838358686710220897394966, −1.19803253808078539724524713519,
1.00939157423620863854624350243, 2.05517397778343680712637215599, 2.96982108910212289889530632582, 4.37711620695467741730742704184, 5.18415674415829578177572666582, 5.79824944836685533999418841146, 7.33107821098916724184006496721, 7.52931264282088521031421367546, 7.891792152543701764797607351628, 9.267357416403957812413522669317
