| L(s) = 1 | − 0.486i·3-s + 3.63i·7-s + 2.76·9-s − 2.79·11-s − 2.86i·13-s + 1.17i·17-s − 19-s + 1.76·21-s + 0.617i·23-s − 2.80i·27-s + 4.96·29-s + 0.745·31-s + 1.36i·33-s + 8.23i·37-s − 1.39·39-s + ⋯ |
| L(s) = 1 | − 0.280i·3-s + 1.37i·7-s + 0.921·9-s − 0.842·11-s − 0.794i·13-s + 0.284i·17-s − 0.229·19-s + 0.385·21-s + 0.128i·23-s − 0.539i·27-s + 0.922·29-s + 0.133·31-s + 0.236i·33-s + 1.35i·37-s − 0.223·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.723632572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.723632572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.486iT - 3T^{2} \) |
| 7 | \( 1 - 3.63iT - 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 2.86iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 23 | \( 1 - 0.617iT - 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 - 0.745T + 31T^{2} \) |
| 37 | \( 1 - 8.23iT - 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 - 5.07iT - 47T^{2} \) |
| 53 | \( 1 - 7.45iT - 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.10iT - 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 9.52iT - 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 4.66iT - 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.629iT - 97T^{2} \) |
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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
−8.505275648169241149557266855510, −7.978399692635477727106517732481, −7.22200227210071217793561255222, −6.33295002879587551071303461953, −5.65840528633260754015762968515, −4.99440419626286979827565832039, −4.07535121098109295322587538063, −2.86076881234757663339605883219, −2.32453926498949642698190028562, −1.09840946982056016654984022510,
0.56813902125953105382665538383, 1.71796565932186849500069916082, 2.87699558615811092931288419089, 4.00289113695906207005620083935, 4.38150076578341967162192774991, 5.17270831461869661792096266459, 6.30496450022956417834707552311, 7.01680772205473941745298691743, 7.53781232072267666847099064965, 8.234325372952187047085370066715
