| L(s) = 1 | − 1.87i·3-s − i·5-s + 2.98·7-s − 0.523·9-s − 0.207·11-s + 1.01·13-s − 1.87·15-s + 0.527i·17-s + 2.62·19-s − 5.60i·21-s + (−4.13 + 2.43i)23-s − 25-s − 4.64i·27-s + 5.03·29-s − 3.25i·31-s + ⋯ |
| L(s) = 1 | − 1.08i·3-s − 0.447i·5-s + 1.12·7-s − 0.174·9-s − 0.0626·11-s + 0.281·13-s − 0.484·15-s + 0.127i·17-s + 0.603·19-s − 1.22i·21-s + (−0.861 + 0.507i)23-s − 0.200·25-s − 0.894i·27-s + 0.934·29-s − 0.584i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293995198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293995198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (4.13 - 2.43i)T \) |
| good | 3 | \( 1 + 1.87iT - 3T^{2} \) |
| 7 | \( 1 - 2.98T + 7T^{2} \) |
| 11 | \( 1 + 0.207T + 11T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 17 | \( 1 - 0.527iT - 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 - 2.23iT - 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 3.66iT - 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 - 4.07iT - 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 7.36iT - 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 - 0.921T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 12.0iT - 89T^{2} \) |
| 97 | \( 1 - 7.54iT - 97T^{2} \) |
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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
−8.075633438046528730466091833817, −7.75794301022821473597295223533, −6.93874113336445237724046852159, −6.13697488848056001300180988259, −5.35436618492123923333304171871, −4.58059326040785556332544628419, −3.69476601768073317370083492127, −2.34337526267040061769351819122, −1.62280670193854620146684899842, −0.76719980189495546654381089032,
1.21958256977736148420637053670, 2.42351495269179752047602005727, 3.40056153875592950009686438033, 4.28870563338931984127121693888, 4.77252747199915921288484863627, 5.60432664139209584677912705759, 6.43884418748169715771750722964, 7.44991463711802499186867180359, 7.987551747979134034633524744994, 8.856313860488789485212287703805
