| L(s) = 1 | + (2 + 2i)5-s + 4.24i·7-s + (2.82 + 2.82i)11-s + (−3 + 3i)13-s + 6·17-s + (1.41 − 1.41i)19-s + 2.82i·23-s + 3i·25-s + (4 − 4i)29-s − 4.24·31-s + (−8.48 + 8.48i)35-s + (−3 − 3i)37-s − 10i·41-s + (4.24 + 4.24i)43-s − 2.82·47-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.894i)5-s + 1.60i·7-s + (0.852 + 0.852i)11-s + (−0.832 + 0.832i)13-s + 1.45·17-s + (0.324 − 0.324i)19-s + 0.589i·23-s + 0.600i·25-s + (0.742 − 0.742i)29-s − 0.762·31-s + (−1.43 + 1.43i)35-s + (−0.493 − 0.493i)37-s − 1.56i·41-s + (0.646 + 0.646i)43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.197679817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197679817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2 - 2i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-4 + 4i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (3 - 3i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.491169331713595172455090448555, −8.701907255520739268137237709666, −7.49382583097022195987952893194, −6.93936862232841455260324500983, −5.98844311392700331368739220852, −5.56274248591002530826125794123, −4.54417497817803933031489401397, −3.27585321719387904688766880416, −2.38267782007280543547009143943, −1.75056685549526493265060082469,
0.808573877999497436620348731660, 1.40427380150709814908647425834, 3.05326194989636563524044788726, 3.85012808496995965918658611942, 4.89216196036654126963001983910, 5.50820553313430614408276083830, 6.40674410564623498435249081113, 7.27355556954059021980891881994, 8.004073417903941993339136082733, 8.762523233871437077244937075107
