| L(s) = 1 | + 2-s + 3-s + 4-s + 1.23·5-s + 6-s + 1.47·7-s + 8-s + 9-s + 1.23·10-s + 0.886·11-s + 12-s + 0.0673·13-s + 1.47·14-s + 1.23·15-s + 16-s + 4.29·17-s + 18-s + 4.66·19-s + 1.23·20-s + 1.47·21-s + 0.886·22-s + 24-s − 3.47·25-s + 0.0673·26-s + 27-s + 1.47·28-s − 7.67·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.552·5-s + 0.408·6-s + 0.558·7-s + 0.353·8-s + 0.333·9-s + 0.391·10-s + 0.267·11-s + 0.288·12-s + 0.0186·13-s + 0.395·14-s + 0.319·15-s + 0.250·16-s + 1.04·17-s + 0.235·18-s + 1.07·19-s + 0.276·20-s + 0.322·21-s + 0.189·22-s + 0.204·24-s − 0.694·25-s + 0.0132·26-s + 0.192·27-s + 0.279·28-s − 1.42·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.668991279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.668991279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 11 | \( 1 - 0.886T + 11T^{2} \) |
| 13 | \( 1 - 0.0673T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 29 | \( 1 + 7.67T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 + 0.135T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 - 8.40T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.660337313572829024323060885005, −7.62859728836470563970920381580, −7.43135460906210555841156849401, −6.20655155852946601283794211654, −5.59033243791346915519517494805, −4.86353222748798789137270623441, −3.86319239526885183766768058481, −3.18702547441187525893421740232, −2.12896001576302614393259142427, −1.30659575721684189850819494350,
1.30659575721684189850819494350, 2.12896001576302614393259142427, 3.18702547441187525893421740232, 3.86319239526885183766768058481, 4.86353222748798789137270623441, 5.59033243791346915519517494805, 6.20655155852946601283794211654, 7.43135460906210555841156849401, 7.62859728836470563970920381580, 8.660337313572829024323060885005
