| L(s) = 1 | − 21.0·5-s + (11 − 14.8i)7-s − 15.5i·11-s − 29.7i·13-s + 63.2·17-s − 89.3i·19-s + 77.7i·23-s + 318.·25-s + 125. i·29-s − 238. i·31-s + (−231. + 313. i)35-s − 184·37-s + 105.·41-s + 190·43-s + 42.1·47-s + ⋯ |
| L(s) = 1 | − 1.88·5-s + (0.593 − 0.804i)7-s − 0.426i·11-s − 0.635i·13-s + 0.901·17-s − 1.07i·19-s + 0.705i·23-s + 2.55·25-s + 0.805i·29-s − 1.38i·31-s + (−1.11 + 1.51i)35-s − 0.817·37-s + 0.401·41-s + 0.673·43-s + 0.130·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0204i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5756288942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5756288942\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-11 + 14.8i)T \) |
| good | 5 | \( 1 + 21.0T + 125T^{2} \) |
| 11 | \( 1 + 15.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 29.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 63.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 77.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 238. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 184T + 5.06e4T^{2} \) |
| 41 | \( 1 - 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 190T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 357. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 84.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 655. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 296T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 804. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 836T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 695.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 566. iT - 9.12e5T^{2} \) |
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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
−8.934139178341758516863075396130, −8.142973053132720858472103604369, −7.53765529368854379669555148368, −7.03303589619805230688352029660, −5.54604821581197858399795919200, −4.57289845668700493157914744029, −3.79775957083581739833083276745, −3.01264545073106386118664557784, −1.06625760258022828552797566299, −0.18062066221630131497171129441,
1.36011212350590107593851550640, 2.82147345830056843226560733987, 3.90237207799357689496829756303, 4.58581311774775998305184631169, 5.60226913390946430457912466678, 6.84240309975801776018506888278, 7.64598220965650189916550685810, 8.274958533798591959796274567763, 8.888929884933184581883646309091, 10.06663803149199881769753788942
