| L(s) = 1 | − 0.367·3-s + (1.82 − 1.29i)5-s + 2.99i·7-s − 2.86·9-s − 1.08i·11-s + 1.37i·13-s + (−0.671 + 0.474i)15-s − 0.212·17-s − 6.70i·19-s − 1.10i·21-s − 1.06i·23-s + (1.67 − 4.71i)25-s + 2.15·27-s + (2.49 + 4.77i)29-s − 4.22i·31-s + ⋯ |
| L(s) = 1 | − 0.212·3-s + (0.816 − 0.576i)5-s + 1.13i·7-s − 0.954·9-s − 0.328i·11-s + 0.381i·13-s + (−0.173 + 0.122i)15-s − 0.0514·17-s − 1.53i·19-s − 0.240i·21-s − 0.221i·23-s + (0.334 − 0.942i)25-s + 0.414·27-s + (0.463 + 0.885i)29-s − 0.759i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730003332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730003332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.82 + 1.29i)T \) |
| 29 | \( 1 + (-2.49 - 4.77i)T \) |
| good | 3 | \( 1 + 0.367T + 3T^{2} \) |
| 7 | \( 1 - 2.99iT - 7T^{2} \) |
| 11 | \( 1 + 1.08iT - 11T^{2} \) |
| 13 | \( 1 - 1.37iT - 13T^{2} \) |
| 17 | \( 1 + 0.212T + 17T^{2} \) |
| 19 | \( 1 + 6.70iT - 19T^{2} \) |
| 23 | \( 1 + 1.06iT - 23T^{2} \) |
| 31 | \( 1 + 4.22iT - 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 + 3.38iT - 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 + 3.51T + 73T^{2} \) |
| 79 | \( 1 - 8.85iT - 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.95iT - 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.941891498751265960253099783054, −8.570072937891955611828954518687, −7.36600889335346990520643877611, −6.34161504927994886190258565720, −5.74410358434502688949487414804, −5.21223588407278888068685388512, −4.27328634480495406295514818932, −2.74833289273959399710371442345, −2.31649144243449649184347188619, −0.74619309900459085431190112885,
1.00177674549965913199340154340, 2.28626412113608997454847022931, 3.25530328949253979997796390827, 4.16233682245354328695249624321, 5.24954181377289184927662779296, 6.02560859918687396833705145282, 6.57693085324072920006352990059, 7.57896380546999407766251411467, 8.103263324142196843406749900446, 9.203627102936649282237827748747
