| L(s) = 1 | + (−0.618 − 1.61i)3-s + (1 + 2i)5-s + 1.23·7-s + (−2.23 + 2.00i)9-s − 4i·11-s − 2.47·13-s + (2.61 − 2.85i)15-s − 2.47·17-s − 6.47·19-s + (−0.763 − 2.00i)21-s + 3.23i·23-s + (−3 + 4i)25-s + (4.61 + 2.38i)27-s + 4.47·29-s − 6.47i·31-s + ⋯ |
| L(s) = 1 | + (−0.356 − 0.934i)3-s + (0.447 + 0.894i)5-s + 0.467·7-s + (−0.745 + 0.666i)9-s − 1.20i·11-s − 0.685·13-s + (0.675 − 0.736i)15-s − 0.599·17-s − 1.48·19-s + (−0.166 − 0.436i)21-s + 0.674i·23-s + (−0.600 + 0.800i)25-s + (0.888 + 0.458i)27-s + 0.830·29-s − 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4061453657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4061453657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 3.23iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47iT - 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 4.76iT - 43T^{2} \) |
| 47 | \( 1 - 4.76iT - 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 1.70iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 1.52iT - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 4.94iT - 97T^{2} \) |
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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
−8.542418756422817487451442002394, −8.066893678839832999664976633191, −7.03766347905728255170193611352, −6.53725994968526653247886971582, −5.78478636799960283712249966719, −4.98896511305014510332638662827, −3.63485291561661281203834427280, −2.52626897735491081239962309484, −1.80235886716696537391861256371, −0.14217652097523387169408254462,
1.63598078590768218044534269461, 2.72732191546101435236077852978, 4.40694483337951729227080035853, 4.50647687105076591454811424145, 5.32778911720168240918052692539, 6.32246079460045972849572291724, 7.11742603178275326594174531458, 8.448051945995949393137130247767, 8.752741266306328039292107641089, 9.655824375165911774615910324279
