| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.722288710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722288710\) |
| \(L(\frac{3}{2})\) |
|
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
−8.284031660811029707422344535508, −7.61118264748322836756009451778, −6.61573641301954895655746054641, −6.24531944677507708821045963480, −4.90815584890649194922573976873, −4.71243422800308752284588939880, −3.81726325776408415036782540117, −2.83123470160447898015171932648, −1.41292534991297318688680207295, −1.03934401927586098198823339308,
1.04501449932362816969001895344, 2.00006168355533106840322368111, 2.90874204157787110977645082566, 4.02851634869491148494060839827, 4.35827639995361522216335862343, 5.70100276846534893960919157106, 6.10021985423526126426274377836, 6.86240341996644203430561721120, 7.65777685508556217751083486401, 8.426277741122887234177918338074
