| L(s) = 1 | − 2i·2-s − 3i·3-s + 4·4-s − 6·6-s − 7i·7-s − 24i·8-s − 9·9-s − 21·11-s − 12i·12-s − 24i·13-s − 14·14-s − 16·16-s − 22i·17-s + 18i·18-s − 16·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s − 0.408·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s − 0.575·11-s − 0.288i·12-s − 0.512i·13-s − 0.267·14-s − 0.250·16-s − 0.313i·17-s + 0.235i·18-s − 0.193·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.382361315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382361315\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 2 | \( 1 + 2iT - 8T^{2} \) |
| 11 | \( 1 + 21T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 22iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 16T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 167T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 400iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 182iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 488T + 2.05e5T^{2} \) |
| 61 | \( 1 - 28T + 2.26e5T^{2} \) |
| 67 | \( 1 + 967iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 285T + 3.57e5T^{2} \) |
| 73 | \( 1 - 838iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 469T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 324T + 7.04e5T^{2} \) |
| 97 | \( 1 + 114iT - 9.12e5T^{2} \) |
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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
−10.21420790210294167924953949616, −9.223980873564047010625797149322, −7.948282652397653849486575767798, −7.28808761962150163831461415376, −6.35826795053767554196357805109, −5.26994691689554823239706844716, −3.77574623727867459499807250738, −2.75288179062054437215414622114, −1.68071640212912531972121827183, −0.38142142708802669271933037256,
1.94354790731588735069840882354, 3.11999090111784363946957797728, 4.54174436188252590321305161765, 5.53096164188063286123335215275, 6.30861724904856560383403885985, 7.33274278909123766119537041455, 8.228796414578987652865381425721, 9.034035468909475440393734547129, 10.09822296234234106787534183967, 10.95434016999755428996981208780
