| L(s) = 1 | − 0.717i·2-s + 1.56i·3-s + 1.48·4-s + (0.811 − 2.08i)5-s + 1.12·6-s − 2.51i·7-s − 2.49i·8-s + 0.535·9-s + (−1.49 − 0.581i)10-s − 5.85·11-s + 2.33i·12-s − 0.791i·13-s − 1.80·14-s + (3.27 + 1.27i)15-s + 1.17·16-s − 0.651i·17-s + ⋯ |
| L(s) = 1 | − 0.507i·2-s + 0.906i·3-s + 0.742·4-s + (0.362 − 0.931i)5-s + 0.459·6-s − 0.952i·7-s − 0.883i·8-s + 0.178·9-s + (−0.472 − 0.183i)10-s − 1.76·11-s + 0.673i·12-s − 0.219i·13-s − 0.482·14-s + (0.844 + 0.328i)15-s + 0.294·16-s − 0.157i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852291028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852291028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.811 + 2.08i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.717iT - 2T^{2} \) |
| 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 7 | \( 1 + 2.51iT - 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 + 0.791iT - 13T^{2} \) |
| 17 | \( 1 + 0.651iT - 17T^{2} \) |
| 23 | \( 1 + 4.88iT - 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 0.741iT - 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 + 0.761iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 6.05T + 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 3.26iT - 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−9.326382454071176323499536048024, −8.236660779519274593713517803219, −7.56622029258066328831517412631, −6.66504696758620456812709859157, −5.53411556051841310633161385357, −4.81773103671980383158846856585, −4.04271204301533250823181228529, −3.03289984542557293077449402670, −1.96209350994045207308065939452, −0.62066247085532964721675995595,
1.80196201287786673139558657242, 2.42116606800860355961030921236, 3.21326508617692645954131783083, 5.03830275387404548409092611783, 5.79241341846132851285305440385, 6.38513792485628329890606601265, 7.15895274198292110982864868846, 7.76133916780733040088729980106, 8.282314249552113211680154836711, 9.568968594564731856827669319764
