| L(s) = 1 | + 20.1i·5-s + 5.50i·7-s − 4.17·11-s + 38.7·13-s + 84.9i·17-s − 140. i·19-s + 36.4·23-s − 279.·25-s + 129. i·29-s + 264. i·31-s − 110.·35-s − 28.1·37-s + 426. i·41-s + 357. i·43-s + 303.·47-s + ⋯ |
| L(s) = 1 | + 1.79i·5-s + 0.297i·7-s − 0.114·11-s + 0.825·13-s + 1.21i·17-s − 1.69i·19-s + 0.330·23-s − 2.23·25-s + 0.830i·29-s + 1.53i·31-s − 0.534·35-s − 0.125·37-s + 1.62i·41-s + 1.26i·43-s + 0.942·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.539534850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539534850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 20.1iT - 125T^{2} \) |
| 7 | \( 1 - 5.50iT - 343T^{2} \) |
| 11 | \( 1 + 4.17T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 140. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 36.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 264. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 28.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 426. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 357. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 101. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 179. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 515.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 448.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.35e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 726.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.338661817562600520203998416588, −8.570597258915168080834013056406, −7.66654720025367962003484055913, −6.76100523633991102942486389830, −6.43624991284646473345949970122, −5.45103550648858363511939673974, −4.25578630506757295811931096018, −3.18938876755027434637432376258, −2.71536377444833269260739659960, −1.42989986216070388833434320479,
0.35486663758090216079874666408, 1.12924290323367850010662438326, 2.21248529997116871698125848554, 3.82419530010863943349981788582, 4.28111488828309675057182078401, 5.48406546671306550623848586666, 5.73526141881497194028260334761, 7.12293699436964246054656409615, 7.964688480708215413912826707159, 8.557902034170972237478897658143
