| L(s) = 1 | + i·3-s + (−1.21 − 1.87i)5-s − 4.68i·7-s − 9-s − 11-s − 4.91i·13-s + (1.87 − 1.21i)15-s + 2.37i·17-s + 8.05·19-s + 4.68·21-s − 2.20i·23-s + (−2.02 + 4.57i)25-s − i·27-s − 0.589·29-s + 5.73·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.545 − 0.838i)5-s − 1.76i·7-s − 0.333·9-s − 0.301·11-s − 1.36i·13-s + (0.483 − 0.314i)15-s + 0.576i·17-s + 1.84·19-s + 1.02·21-s − 0.459i·23-s + (−0.405 + 0.914i)25-s − 0.192i·27-s − 0.109·29-s + 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059545536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059545536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.21 + 1.87i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4.68iT - 7T^{2} \) |
| 13 | \( 1 + 4.91iT - 13T^{2} \) |
| 17 | \( 1 - 2.37iT - 17T^{2} \) |
| 19 | \( 1 - 8.05T + 19T^{2} \) |
| 23 | \( 1 + 2.20iT - 23T^{2} \) |
| 29 | \( 1 + 0.589T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 - 3.31iT - 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 0.611iT - 43T^{2} \) |
| 47 | \( 1 + 8.03iT - 47T^{2} \) |
| 53 | \( 1 + 4.66iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 - 5.74iT - 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 7.81iT - 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 6.32iT - 97T^{2} \) |
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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
−8.383827394268422636328722718673, −7.82644434516346674718331009526, −7.31753902293657003104730033349, −6.17928785188620915617932451109, −5.08500009604340796004289812870, −4.70369327614267719346726819771, −3.64613677656039660191607369986, −3.20944303465586944088511574156, −1.26945968088213244629777371608, −0.37527972889381623473186802309,
1.60286947902826677893943851124, 2.68784838613361197663546189559, 3.13775410748702553511156110670, 4.51016833104766243176973754050, 5.45899804796749875562727580558, 6.14466489309304495747507300397, 6.93241438585142250124768895333, 7.60979149414205184791802421246, 8.312642252164253599270468341468, 9.275859492141202729362838443630
