| L(s) = 1 | + (−1.17 + 0.794i)2-s + (0.738 − 1.85i)4-s − i·5-s + 2.32i·7-s + (0.611 + 2.76i)8-s + (0.794 + 1.17i)10-s + 2.36·11-s + 13-s + (−1.84 − 2.72i)14-s + (−2.90 − 2.74i)16-s − 5.20i·17-s + 6.04i·19-s + (−1.85 − 0.738i)20-s + (−2.76 + 1.87i)22-s − 4.93·23-s + ⋯ |
| L(s) = 1 | + (−0.827 + 0.561i)2-s + (0.369 − 0.929i)4-s − 0.447i·5-s + 0.880i·7-s + (0.216 + 0.976i)8-s + (0.251 + 0.370i)10-s + 0.713·11-s + 0.277·13-s + (−0.494 − 0.728i)14-s + (−0.727 − 0.686i)16-s − 1.26i·17-s + 1.38i·19-s + (−0.415 − 0.165i)20-s + (−0.590 + 0.400i)22-s − 1.02·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9911863726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9911863726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 - 0.794i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.32iT - 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 17 | \( 1 + 5.20iT - 17T^{2} \) |
| 19 | \( 1 - 6.04iT - 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 3.63iT - 29T^{2} \) |
| 31 | \( 1 - 8.68iT - 31T^{2} \) |
| 37 | \( 1 - 2.01T + 37T^{2} \) |
| 41 | \( 1 - 4.97iT - 41T^{2} \) |
| 43 | \( 1 - 4.21iT - 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 + 13.1iT - 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + 7.42iT - 67T^{2} \) |
| 71 | \( 1 - 8.91T + 71T^{2} \) |
| 73 | \( 1 + 0.580T + 73T^{2} \) |
| 79 | \( 1 - 9.28iT - 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 - 5.84iT - 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−9.224135501772208939426990144118, −8.402604088998042965759280425632, −7.918260258825662299840596435335, −6.88642357305418777670116019375, −6.19472416299740599528401655652, −5.45726011836656257945536431281, −4.71722782186828185383663647689, −3.40521068190802073508513174955, −2.12859161361392270036809755455, −1.16245902022351435540168076616,
0.49777135799946917334308264607, 1.73591532501399758080235671321, 2.74365263778216579566513246762, 3.97637828160858671438572579002, 4.17510236158070743208561854877, 5.94888376895905501214843626663, 6.60173341968593499662422486493, 7.44259267464959612460256651691, 7.970019968641686975818504190080, 8.938716061186940314238235348150
