L(s) = 1 | + (−1.56 − 2.70i)5-s + (17.1 − 6.88i)7-s + (33.2 + 19.2i)11-s + 13.8i·13-s + (−47.5 + 82.4i)17-s + (9.86 − 5.69i)19-s + (23.9 − 13.8i)23-s + (57.6 − 99.7i)25-s − 44.2i·29-s + (−119. − 68.7i)31-s + (−45.4 − 35.7i)35-s + (70.6 + 122. i)37-s + 337.·41-s + 417.·43-s + (145. + 251. i)47-s + ⋯ |
L(s) = 1 | + (−0.139 − 0.242i)5-s + (0.928 − 0.371i)7-s + (0.912 + 0.526i)11-s + 0.294i·13-s + (−0.678 + 1.17i)17-s + (0.119 − 0.0687i)19-s + (0.217 − 0.125i)23-s + (0.460 − 0.798i)25-s − 0.283i·29-s + (−0.690 − 0.398i)31-s + (−0.219 − 0.172i)35-s + (0.313 + 0.543i)37-s + 1.28·41-s + 1.48·43-s + (0.450 + 0.779i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.237486389\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.237486389\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-17.1 + 6.88i)T \) |
good | 5 | \( 1 + (1.56 + 2.70i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-33.2 - 19.2i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 13.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (47.5 - 82.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-9.86 + 5.69i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-23.9 + 13.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 44.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (119. + 68.7i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-70.6 - 122. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-145. - 251. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (14.7 + 8.53i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (299. - 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-459. + 265. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-325. + 563. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 934. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-787. - 454. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (397. + 688. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 314.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-179. - 310. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 80.5iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.70473124900923016384336846351, −9.541812801372215699837996112096, −8.717216350870893189872805265802, −7.86116489569989822129386138831, −6.90631028391519219496157662035, −5.90373479921296156657630167212, −4.53425881519571596129073551201, −4.03870793861668848659657037504, −2.21632915276568014050766557021, −1.03898276460513388204614974726,
0.926702944733652762310607492056, 2.37277446195190318159171857325, 3.65413981621176127171894148691, 4.84201424516975010696387138859, 5.75961615076726083027118917677, 6.94460331607877295118164012830, 7.73232533208892094126600104176, 8.883000919946239433311830276639, 9.337649797073972603102518330856, 10.83635085634328437088345781640