| L(s) = 1 | + 5-s − 3.53i·11-s + 0.599i·13-s − 1.69·17-s + 7.78i·19-s + 3.31i·23-s + 25-s − 0.456i·29-s − 1.05i·31-s − 0.386·37-s − 0.478·41-s − 4.21·43-s + 8.55·47-s − 3.98i·53-s − 3.53i·55-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.06i·11-s + 0.166i·13-s − 0.411·17-s + 1.78i·19-s + 0.690i·23-s + 0.200·25-s − 0.0846i·29-s − 0.189i·31-s − 0.0634·37-s − 0.0746·41-s − 0.642·43-s + 1.24·47-s − 0.548i·53-s − 0.476i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420599185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420599185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 13 | \( 1 - 0.599iT - 13T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 - 7.78iT - 19T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 0.456iT - 29T^{2} \) |
| 31 | \( 1 + 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 0.386T + 37T^{2} \) |
| 41 | \( 1 + 0.478T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 + 3.98iT - 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 2.87T + 67T^{2} \) |
| 71 | \( 1 - 0.336iT - 71T^{2} \) |
| 73 | \( 1 - 3.08iT - 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.18iT - 97T^{2} \) |
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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
−7.985237515439748228566129212412, −7.26614335909480946847443871955, −6.46667632791737970318336215250, −5.76618442229039648167088288208, −5.46549556250690458584676191966, −4.32910746713529930453544927568, −3.68038433251132305266593237813, −2.89978090384708892414216546168, −1.92766250717594621733930181496, −1.08769706850272936544623876378,
0.32991635147575727100624257857, 1.58285896488390651647377925877, 2.42853395668153177389814621405, 3.05945196313703571489907908898, 4.28850789460348533405934167430, 4.70008313602996172660480995564, 5.45485728319196776994556708020, 6.27753178927883734709831123060, 7.03091945187672989797659424834, 7.28425761249121261562967029898
