| L(s) = 1 | − 3-s − 5-s − 2.77·7-s + 9-s − 0.778·11-s − 5.50·13-s + 15-s − 7.00·17-s + 19-s + 2.77·21-s − 1.27·23-s + 25-s − 27-s + 8.23·29-s + 5.00·31-s + 0.778·33-s + 2.77·35-s + 8.51·37-s + 5.50·39-s + 9.06·41-s − 10.5·43-s − 45-s − 6.72·47-s + 0.717·49-s + 7.00·51-s + 8.72·53-s + 0.778·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.04·7-s + 0.333·9-s − 0.234·11-s − 1.52·13-s + 0.258·15-s − 1.70·17-s + 0.229·19-s + 0.606·21-s − 0.265·23-s + 0.200·25-s − 0.192·27-s + 1.52·29-s + 0.899·31-s + 0.135·33-s + 0.469·35-s + 1.39·37-s + 0.881·39-s + 1.41·41-s − 1.60·43-s − 0.149·45-s − 0.981·47-s + 0.102·49-s + 0.981·51-s + 1.19·53-s + 0.104·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6394945861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6394945861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 0.778T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 + 7.00T + 17T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 5.00T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 - 9.06T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 + 7.73T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 2.28T + 79T^{2} \) |
| 83 | \( 1 - 3.45T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
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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
−9.143530725068651964054650001054, −8.179964310081354994672688224658, −7.34330198014524915459169362120, −6.61728795118162019768568763515, −6.09810116047250525231431237307, −4.79551937360616566535910273697, −4.46336391010847641818491404612, −3.13419697385842829772938121604, −2.32292200278481330769654759971, −0.50219645841858026346765490631,
0.50219645841858026346765490631, 2.32292200278481330769654759971, 3.13419697385842829772938121604, 4.46336391010847641818491404612, 4.79551937360616566535910273697, 6.09810116047250525231431237307, 6.61728795118162019768568763515, 7.34330198014524915459169362120, 8.179964310081354994672688224658, 9.143530725068651964054650001054
