| L(s) = 1 | − i·3-s + (−1.52 − 1.63i)5-s + 2.82i·7-s − 9-s + 1.05·11-s + i·13-s + (−1.63 + 1.52i)15-s − 7.14i·17-s + 6.61·19-s + 2.82·21-s − 3.49i·23-s + (−0.332 + 4.98i)25-s + i·27-s − 4.82·29-s − 9.65·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.683 − 0.730i)5-s + 1.06i·7-s − 0.333·9-s + 0.318·11-s + 0.277i·13-s + (−0.421 + 0.394i)15-s − 1.73i·17-s + 1.51·19-s + 0.617·21-s − 0.728i·23-s + (−0.0664 + 0.997i)25-s + 0.192i·27-s − 0.896·29-s − 1.73·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9550322077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9550322077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.52 + 1.63i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 17 | \( 1 + 7.14iT - 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.49iT - 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 + 8.11iT - 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 1.46iT - 67T^{2} \) |
| 71 | \( 1 - 0.845T + 71T^{2} \) |
| 73 | \( 1 - 9.28iT - 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 + 0.139T + 89T^{2} \) |
| 97 | \( 1 - 6.93iT - 97T^{2} \) |
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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
−9.156104568986603209302720219421, −8.425710946399664626886374480693, −7.37007231997022738631708572556, −7.04954840687426190621186873936, −5.48064871745384040468291558309, −5.35596705605047348719435194932, −4.00893366238879081190730047983, −2.95070630284712838928064241125, −1.80870553736570180043334892549, −0.38663900231344167256825550283,
1.44414069329534204352416202408, 3.24556806492932409876295898900, 3.66406695562460104175869117850, 4.52569563168075717300980693446, 5.67388541697200259969377844397, 6.55499124339158334242006921669, 7.56611576862048301811739030771, 7.85919575176183854297966279715, 9.052056770082278679883030918496, 9.882398808848400501527937630329
