| L(s) = 1 | − 1.60i·5-s + (2.41 − 1.08i)7-s + 4.20·11-s − 3.26·13-s − 0.666i·17-s + (2.31 − 3.69i)19-s + 1.74·23-s + 2.41·25-s − 8.97i·29-s − 7.89·31-s + (−1.74 − 3.88i)35-s − 8.54i·37-s − 9.71·41-s − 0.242·43-s + 11.6i·47-s + ⋯ |
| L(s) = 1 | − 0.719i·5-s + (0.912 − 0.409i)7-s + 1.26·11-s − 0.906·13-s − 0.161i·17-s + (0.530 − 0.847i)19-s + 0.362·23-s + 0.482·25-s − 1.66i·29-s − 1.41·31-s + (−0.294 − 0.656i)35-s − 1.40i·37-s − 1.51·41-s − 0.0370·43-s + 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095031377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095031377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.41 + 1.08i)T \) |
| 19 | \( 1 + (-2.31 + 3.69i)T \) |
| good | 5 | \( 1 + 1.60iT - 5T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 + 0.666iT - 17T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 + 8.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + 8.54iT - 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 3.71iT - 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 1.53iT - 61T^{2} \) |
| 67 | \( 1 - 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 3.71iT - 71T^{2} \) |
| 73 | \( 1 + 9.81iT - 73T^{2} \) |
| 79 | \( 1 + 8.54iT - 79T^{2} \) |
| 83 | \( 1 - 4.82iT - 83T^{2} \) |
| 89 | \( 1 - 9.71T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.069982485897727966609533492286, −7.33409881637125954661676123523, −6.85285961939744373020054794707, −5.75547120305094020926607521217, −5.03141573683779120916248237055, −4.43907092647163097300500084571, −3.73359178085454190835503285441, −2.49108137568174012445207095885, −1.51579455773804317459640423321, −0.59668871500431042970293305252,
1.32781473384584977593737571506, 2.07933106303448022935756249842, 3.22432034334378853842574584539, 3.82090719900152820138368880635, 5.03721008842534788265494006787, 5.32318154883214199847780431688, 6.55836585335158008566916095701, 6.93992908176818501172474288876, 7.68358981211229583946080838017, 8.579286094498864431733515400296
