L(s) = 1 | + 3i·3-s + 1.28i·5-s − 7·7-s − 9·9-s + 58.8i·11-s + 11.2i·13-s − 3.84·15-s − 104.·17-s + 51.5i·19-s − 21i·21-s − 131.·23-s + 123.·25-s − 27i·27-s − 32.6i·29-s + 108.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.114i·5-s − 0.377·7-s − 0.333·9-s + 1.61i·11-s + 0.240i·13-s − 0.0661·15-s − 1.48·17-s + 0.621i·19-s − 0.218i·21-s − 1.19·23-s + 0.986·25-s − 0.192i·27-s − 0.209i·29-s + 0.630·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.03709095511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03709095511\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + 7T \) |
good | 5 | \( 1 - 1.28iT - 125T^{2} \) |
| 11 | \( 1 - 58.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 11.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 51.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 32.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.75iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 422.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 209. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 74.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 447. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 328. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 148.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 35.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 393. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 480.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−9.969428335891032232783007911697, −9.080670597953063197478285492047, −8.355084567397393315429293837396, −7.22702442456112355642146912033, −6.63604458554148933716963689737, −5.62659643207631325722546594874, −4.50486083046235500808888881500, −4.08677583571262674415245455247, −2.72787719285774956850502406753, −1.80611811223100887958736839281,
0.009516082080128935397374884144, 0.958813618390841969968366924030, 2.37424946131615686190509888410, 3.22616750366532182779441421009, 4.34641191793435695093873767831, 5.47428575665323497960994480515, 6.30238955506845362639937996022, 6.86100624237400654201469733217, 7.985461305202341088803718048790, 8.644876544113581400256586384195