| L(s) = 1 | − 0.540i·3-s + (2.21 − 0.303i)5-s − 1.15i·7-s + 2.70·9-s − 2.10·11-s − 5.59i·13-s + (−0.164 − 1.19i)15-s + 0.244i·17-s + 1.45·19-s − 0.625·21-s − i·23-s + (4.81 − 1.34i)25-s − 3.08i·27-s − 4.29·29-s − 3.19·31-s + ⋯ |
| L(s) = 1 | − 0.312i·3-s + (0.990 − 0.135i)5-s − 0.437i·7-s + 0.902·9-s − 0.635·11-s − 1.55i·13-s + (−0.0423 − 0.309i)15-s + 0.0592i·17-s + 0.332·19-s − 0.136·21-s − 0.208i·23-s + (0.963 − 0.269i)25-s − 0.593i·27-s − 0.797·29-s − 0.574·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.062872690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062872690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 + 0.303i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.540iT - 3T^{2} \) |
| 7 | \( 1 + 1.15iT - 7T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 + 5.59iT - 13T^{2} \) |
| 17 | \( 1 - 0.244iT - 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 0.807iT - 37T^{2} \) |
| 41 | \( 1 + 1.59T + 41T^{2} \) |
| 43 | \( 1 + 5.98iT - 43T^{2} \) |
| 47 | \( 1 - 0.624iT - 47T^{2} \) |
| 53 | \( 1 - 0.536iT - 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 4.61iT - 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 - 8.87iT - 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 6.11iT - 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 + 2.38iT - 97T^{2} \) |
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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
−9.148069473511410761437081290561, −8.182828128282348476158737790238, −7.45973648984336400032796825174, −6.79334025751983249377428526225, −5.69365698949960894634762598979, −5.25546962787634251355930463612, −4.09391308731076453135098419710, −2.96375974017171664087904043027, −1.92258631849996744825699596373, −0.77480310983751863817741400000,
1.55581077336916694610455189530, 2.34348256476272927445057794492, 3.58344375997797571762369662346, 4.62995425696192501857198026988, 5.33449290448384281968820860738, 6.25034540205005842694267474400, 6.98253298692369924995472489909, 7.77641846417529695057583230239, 9.070469608211801423071154633666, 9.304251532327040782159559136059
