| L(s) = 1 | + (2.69 − 1.32i)3-s + 0.640i·5-s − 2.72·7-s + (5.47 − 7.14i)9-s − 11.2i·11-s + 5.25·13-s + (0.849 + 1.72i)15-s − 14.8i·17-s + 15.0·19-s + (−7.31 + 3.61i)21-s − 36.4i·23-s + 24.5·25-s + (5.24 − 26.4i)27-s + 51.7i·29-s + 36.5·31-s + ⋯ |
| L(s) = 1 | + (0.896 − 0.442i)3-s + 0.128i·5-s − 0.388·7-s + (0.608 − 0.793i)9-s − 1.02i·11-s + 0.403·13-s + (0.0566 + 0.114i)15-s − 0.874i·17-s + 0.793·19-s + (−0.348 + 0.171i)21-s − 1.58i·23-s + 0.983·25-s + (0.194 − 0.980i)27-s + 1.78i·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.88551 - 1.17218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88551 - 1.17218i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.69 + 1.32i)T \) |
| good | 5 | \( 1 - 0.640iT - 25T^{2} \) |
| 7 | \( 1 + 2.72T + 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 5.25T + 169T^{2} \) |
| 17 | \( 1 + 14.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.0T + 361T^{2} \) |
| 23 | \( 1 + 36.4iT - 529T^{2} \) |
| 29 | \( 1 - 51.7iT - 841T^{2} \) |
| 31 | \( 1 - 36.5T + 961T^{2} \) |
| 37 | \( 1 + 63.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 61.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 59.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 37.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 58.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 23.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7.29iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 32.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 112. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80.0T + 9.40e3T^{2} \) |
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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
−10.89310034521095932098081993663, −9.975937179726617389736499235955, −8.836901728281994288036274027867, −8.431814869178965608444452464636, −7.10956094446600496689239181750, −6.48426236013114577935469086478, −5.05014472414565936851569421789, −3.50181635712018353166146597611, −2.75136326761587394029992012940, −0.967947560833860947786823882329,
1.72378244215110354799361106732, 3.15086397439557809777194827379, 4.15883191078827555893645497124, 5.28205331174709571984336138200, 6.68467847178147120078245871069, 7.71739069994351970788887373884, 8.532099957434127894818738907895, 9.645827601970612309548568967605, 9.994563390652665544753657982157, 11.18787738683558555794628623323
