L(s) = 1 | + (5.74 + 9.94i)5-s + (−17.2 − 6.73i)7-s + (−49.6 − 28.6i)11-s + 9.20i·13-s + (14.5 − 25.1i)17-s + (32.6 − 18.8i)19-s + (7.27 − 4.19i)23-s + (−3.41 + 5.91i)25-s − 62.3i·29-s + (48.8 + 28.2i)31-s + (−32.1 − 210. i)35-s + (146. + 252. i)37-s + 54.2·41-s + 438.·43-s + (128. + 222. i)47-s + ⋯ |
L(s) = 1 | + (0.513 + 0.889i)5-s + (−0.931 − 0.363i)7-s + (−1.36 − 0.785i)11-s + 0.196i·13-s + (0.207 − 0.359i)17-s + (0.393 − 0.227i)19-s + (0.0659 − 0.0380i)23-s + (−0.0273 + 0.0473i)25-s − 0.399i·29-s + (0.283 + 0.163i)31-s + (−0.155 − 1.01i)35-s + (0.649 + 1.12i)37-s + 0.206·41-s + 1.55·43-s + (0.398 + 0.690i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.612393367\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612393367\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (17.2 + 6.73i)T \) |
good | 5 | \( 1 + (-5.74 - 9.94i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (49.6 + 28.6i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 9.20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-14.5 + 25.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-32.6 + 18.8i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-7.27 + 4.19i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 62.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-48.8 - 28.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-146. - 252. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 54.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-128. - 222. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-515. - 297. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (238. - 412. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (548. - 316. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-308. + 534. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 396. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-39.8 - 23.0i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (344. + 597. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-595. - 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 946. iT - 9.12e5T^{2} \) |
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
−9.829873336438242235005940196929, −8.985849123308330691290348394437, −7.84487520596530390907144777799, −7.14732253788056678609288823592, −6.21646495190686978425563512115, −5.61937908389436244685418114887, −4.33148041440421658497669128660, −2.98138655212733564562361192410, −2.66010737314629875193537955264, −0.77876187947635142697833357725,
0.55774948660893897029286078165, 1.99922509970417993163972787514, 2.97991912927527820339213195770, 4.26244006705662054523855862815, 5.37633729364096988667483979704, 5.76424868246992848743737749575, 7.01683141940524089907685278490, 7.84595343492860090530604213368, 8.790355980097552482723005528013, 9.537941473046269703734499660842