L(s) = 1 | + (0.658 − 1.38i)2-s + (−1.90 − 1.76i)3-s + (−0.228 − 0.279i)4-s + (−2.23 + 0.0902i)5-s + (−3.70 + 1.49i)6-s + (3.79 + 1.09i)7-s + (2.44 − 0.602i)8-s + (0.301 + 3.73i)9-s + (−1.34 + 3.16i)10-s + (1.51 + 1.78i)11-s + (−0.0566 + 0.936i)12-s + (−0.214 + 3.59i)13-s + (4.02 − 4.54i)14-s + (4.42 + 3.76i)15-s + (0.918 − 4.49i)16-s + (2.92 + 5.30i)17-s + ⋯ |
L(s) = 1 | + (0.465 − 0.981i)2-s + (−1.10 − 1.01i)3-s + (−0.114 − 0.139i)4-s + (−0.999 + 0.0403i)5-s + (−1.51 + 0.608i)6-s + (1.43 + 0.415i)7-s + (0.864 − 0.213i)8-s + (0.100 + 1.24i)9-s + (−0.425 + 0.999i)10-s + (0.458 + 0.538i)11-s + (−0.0163 + 0.270i)12-s + (−0.0595 + 0.998i)13-s + (1.07 − 1.21i)14-s + (1.14 + 0.971i)15-s + (0.229 − 1.12i)16-s + (0.708 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29798 - 0.857117i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29798 - 0.857117i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.23 - 0.0902i)T \) |
| 13 | \( 1 + (0.214 - 3.59i)T \) |
good | 2 | \( 1 + (-0.658 + 1.38i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (1.90 + 1.76i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-3.79 - 1.09i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 1.78i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.92 - 5.30i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-3.29 + 0.881i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.740 - 2.76i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 0.761i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (6.93 + 5.43i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-0.303 - 0.404i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (-5.80 - 5.35i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.70 + 11.9i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (4.44 - 1.68i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (5.32 - 8.80i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (2.60 + 1.72i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (0.572 - 0.596i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 1.61i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (0.480 - 2.13i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (-7.66 + 11.1i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-16.2 + 6.15i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-3.43 - 6.53i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (1.22 + 0.329i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (14.5 - 4.84i)T + (77.5 - 58.2i)T^{2} \) |
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
−10.74020902891043112485790951605, −9.293408018420885403609394323268, −7.900052681646565584919064189985, −7.59504532175951068230287582484, −6.63321449810111735989764782213, −5.42312567835594174279243347396, −4.56051941964372982076448500207, −3.68436333415241209754724563588, −1.97782933667236277337184343370, −1.30419824663470859688521963001,
0.934179041197471118148638098134, 3.48910462876153872909155396162, 4.53836931385390930952182697480, 5.07128916261325582890935336630, 5.63604673484822745880739741172, 6.84631421421002782820213724383, 7.71651173204547426136306888076, 8.253248454603379730554903076916, 9.653670076341953230682693499617, 10.69805499456621806287963103384