| L(s) = 1 | − 0.328i·2-s + 1.89·4-s + (2.20 + 0.364i)5-s − 1.86i·7-s − 1.27i·8-s + (0.119 − 0.725i)10-s − 0.564·11-s + 4.95i·13-s − 0.613·14-s + 3.36·16-s + 7.06i·17-s + 3.32·19-s + (4.17 + 0.689i)20-s + 0.185i·22-s − 2.36i·23-s + ⋯ |
| L(s) = 1 | − 0.232i·2-s + 0.945·4-s + (0.986 + 0.163i)5-s − 0.705i·7-s − 0.452i·8-s + (0.0379 − 0.229i)10-s − 0.170·11-s + 1.37i·13-s − 0.164·14-s + 0.840·16-s + 1.71i·17-s + 0.763·19-s + (0.933 + 0.154i)20-s + 0.0395i·22-s − 0.493i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.580020334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580020334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.20 - 0.364i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.328iT - 2T^{2} \) |
| 7 | \( 1 + 1.86iT - 7T^{2} \) |
| 11 | \( 1 + 0.564T + 11T^{2} \) |
| 13 | \( 1 - 4.95iT - 13T^{2} \) |
| 17 | \( 1 - 7.06iT - 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 2.36iT - 23T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 + 8.50iT - 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 4.39iT - 43T^{2} \) |
| 47 | \( 1 - 7.06iT - 47T^{2} \) |
| 53 | \( 1 + 4.18iT - 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 6.61iT - 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 5.77iT - 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 9.41iT - 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 - 19.2iT - 97T^{2} \) |
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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
−9.812316899587306771194329032244, −8.968369329439060919996583499993, −7.894578667589187309136261267902, −6.97298248738833121602425508007, −6.42238173241568847178489936317, −5.66191608925406249420314754368, −4.37234690674957368673166980872, −3.37794180613399952389309464907, −2.17120930831220949092096380969, −1.42584367192593605363377023856,
1.25547384090140798996355541467, 2.62504634502053370994945099204, 3.03756626432552830656904237798, 5.08687450140213464302190776082, 5.44117886918442175383439779127, 6.30920615341992676833502172420, 7.15985396902686149115912821191, 7.976982975918176662026522984662, 8.835969814240682831658051764611, 9.845551621232178779762416393270
