L(s) = 1 | + (−2.05 + 1.18i)2-s + (−0.0643 − 0.0371i)3-s + (1.82 − 3.15i)4-s + 0.176·6-s + (3.20 + 1.84i)7-s + 3.90i·8-s + (−1.49 − 2.59i)9-s − 1.25·11-s + (−0.234 + 0.135i)12-s + (4.24 + 2.44i)13-s − 8.78·14-s + (−0.998 − 1.73i)16-s + (−5.02 + 2.90i)17-s + (6.16 + 3.55i)18-s + (−0.0552 + 0.0957i)19-s + ⋯ |
L(s) = 1 | + (−1.45 + 0.840i)2-s + (−0.0371 − 0.0214i)3-s + (0.911 − 1.57i)4-s + 0.0720·6-s + (1.21 + 0.698i)7-s + 1.38i·8-s + (−0.499 − 0.864i)9-s − 0.378·11-s + (−0.0676 + 0.0390i)12-s + (1.17 + 0.679i)13-s − 2.34·14-s + (−0.249 − 0.432i)16-s + (−1.21 + 0.703i)17-s + (1.45 + 0.838i)18-s + (−0.0126 + 0.0219i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.602694 + 0.525288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602694 + 0.525288i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 + (2.66 - 5.46i)T \) |
good | 2 | \( 1 + (2.05 - 1.18i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.0643 + 0.0371i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.20 - 1.84i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 1.25T + 11T^{2} \) |
| 13 | \( 1 + (-4.24 - 2.44i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.02 - 2.90i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0552 - 0.0957i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.96iT - 23T^{2} \) |
| 29 | \( 1 - 9.26T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 41 | \( 1 + (-2.81 + 4.86i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 0.220iT - 43T^{2} \) |
| 47 | \( 1 - 0.197iT - 47T^{2} \) |
| 53 | \( 1 + (-6.23 + 3.59i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.77 - 9.99i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 2.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.7 - 7.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.33 + 5.78i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.55iT - 73T^{2} \) |
| 79 | \( 1 + (6.15 - 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.49 - 4.90i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.02 - 8.70i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.78iT - 97T^{2} \) |
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
−10.09600067482174540937969602972, −8.976501446005639879152409019882, −8.498432536469164050978821825943, −8.243325323382623028154907446324, −6.80713276715389935141810572933, −6.35699452850146704069081753588, −5.40142473822017421666411721827, −4.13836214676897739543862506563, −2.33125254590356583462478216581, −1.08848121227594146627432303721,
0.77558151818361216161577956765, 1.97048219412522010721393803867, 2.98688186965585279516298190068, 4.41935851586381913208732057254, 5.42603275809519085860996442026, 6.89336396749481985807965111493, 7.88109619670594241948017405744, 8.270162134034417820912955746420, 8.945005495760515157743803163268, 10.11891317807277210770104096341