| L(s) = 1 | − 7i·3-s + 6i·7-s − 22·9-s + 43·11-s − 28i·13-s − 91i·17-s − 35·19-s + 42·21-s − 162i·23-s − 35i·27-s − 160·29-s − 42·31-s − 301i·33-s + 314i·37-s − 196·39-s + ⋯ |
| L(s) = 1 | − 1.34i·3-s + 0.323i·7-s − 0.814·9-s + 1.17·11-s − 0.597i·13-s − 1.29i·17-s − 0.422·19-s + 0.436·21-s − 1.46i·23-s − 0.249i·27-s − 1.02·29-s − 0.243·31-s − 1.58i·33-s + 1.39i·37-s − 0.804·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.546792950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546792950\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 7iT - 27T^{2} \) |
| 7 | \( 1 - 6iT - 343T^{2} \) |
| 11 | \( 1 - 43T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 91iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 35T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 160T + 2.43e4T^{2} \) |
| 31 | \( 1 + 42T + 2.97e4T^{2} \) |
| 37 | \( 1 - 314iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 203T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 196iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 82iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 280T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 412T + 3.57e5T^{2} \) |
| 73 | \( 1 + 763iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 510T + 4.93e5T^{2} \) |
| 83 | \( 1 + 777iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 945T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3iT - 9.12e5T^{2} \) |
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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
−10.58124976652088098601141289543, −9.356631225121501439067614207365, −8.524008431174238540375117713795, −7.52331073564011179322902942975, −6.73173794090874217727267818602, −5.96207732822383775842538285837, −4.58604461128653227671891498493, −2.99649552687594516601490132911, −1.77430323532526187344602828642, −0.52430139752224847275078268789,
1.67333006552176801852205846904, 3.72714629973999664474696684080, 4.00718254765487309133302652114, 5.30984016629710464406361477787, 6.38556181494375497412423532020, 7.55290888388414223994073033392, 8.916328029373309850884494876079, 9.367365767009723180605233287437, 10.33878462683401464361002083989, 11.04800686206738586350046521188
