| L(s) = 1 | + 3.16·3-s + 2.23i·5-s + 4.24i·7-s + 7.00·9-s + 7.07i·15-s + 13.4i·21-s + (−4.74 − 0.707i)23-s − 5.00·25-s + 12.6·27-s − 6·29-s − 9.48·35-s + 12·41-s − 12.7i·43-s + 15.6i·45-s + 9.48·47-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.999i·5-s + 1.60i·7-s + 2.33·9-s + 1.82i·15-s + 2.92i·21-s + (−0.989 − 0.147i)23-s − 1.00·25-s + 2.43·27-s − 1.11·29-s − 1.60·35-s + 1.87·41-s − 1.94i·43-s + 2.33i·45-s + 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.312105968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.312105968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (4.74 + 0.707i)T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.215403497678866252495955612694, −8.770174630115810353971532412091, −7.892227914853925266771239993353, −7.37557812733734881293790762204, −6.33830228616134470610451434397, −5.49144701961845764894916238254, −4.07293377587954730748701419088, −3.36165500083336533606413305160, −2.35789569964417640388409082685, −2.12864841305706741144718164091,
0.992719158316320777705041565287, 1.98267923644825071147193256303, 3.20672562077088951890911286649, 4.22353258160984330676185182317, 4.32511251589903727092955923506, 5.87828133552865796052554823830, 7.19058096652757856726012202484, 7.66981167545775576475447695974, 8.206154656231066002290505884728, 9.105697671966544422037816504857
