| L(s) = 1 | + (−0.879 + 1.10i)2-s + (−0.453 − 1.94i)4-s + i·5-s + 1.56i·7-s + (2.55 + 1.21i)8-s + (−1.10 − 0.879i)10-s + 1.65·11-s − 13-s + (−1.73 − 1.37i)14-s + (−3.58 + 1.76i)16-s − 5.13i·17-s + 5.90i·19-s + (1.94 − 0.453i)20-s + (−1.45 + 1.83i)22-s + 6.28·23-s + ⋯ |
| L(s) = 1 | + (−0.621 + 0.783i)2-s + (−0.226 − 0.973i)4-s + 0.447i·5-s + 0.590i·7-s + (0.903 + 0.427i)8-s + (−0.350 − 0.278i)10-s + 0.499·11-s − 0.277·13-s + (−0.462 − 0.367i)14-s + (−0.897 + 0.441i)16-s − 1.24i·17-s + 1.35i·19-s + (0.435 − 0.101i)20-s + (−0.310 + 0.390i)22-s + 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056683016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056683016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.879 - 1.10i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 1.56iT - 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 17 | \( 1 + 5.13iT - 17T^{2} \) |
| 19 | \( 1 - 5.90iT - 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 - 7.68iT - 29T^{2} \) |
| 31 | \( 1 + 0.700iT - 31T^{2} \) |
| 37 | \( 1 + 7.59T + 37T^{2} \) |
| 41 | \( 1 + 11.6iT - 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 9.80T + 59T^{2} \) |
| 61 | \( 1 + 0.593T + 61T^{2} \) |
| 67 | \( 1 - 2.24iT - 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 + 13.0iT - 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 9.12iT - 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.930754219173524312266381130283, −8.810378715831656279645775546307, −7.49024982967092064680692869082, −7.18449255598060173183766735523, −6.28657473058622203064163311614, −5.49336224587377500446363496774, −4.83535865140869498239435550677, −3.56993023617965357624642588282, −2.42474622159711044372792248685, −1.20135520656026028495581131780,
0.50726665882483842477024028385, 1.56877650437128433451165973385, 2.68370727119054328668472474864, 3.76770423964158470994924110684, 4.40908267953123757953149223741, 5.35107650945931512782668540613, 6.71427779522099957754061686840, 7.18145112841367176133654503213, 8.270403534785971467191713100701, 8.680494912962884539440192580444
