L(s) = 1 | + (1.41 − 0.0591i)2-s + 3-s + (1.99 − 0.167i)4-s + (1.41 − 0.0591i)6-s − 1.33i·7-s + (2.80 − 0.353i)8-s + 9-s − 2.94i·11-s + (1.99 − 0.167i)12-s − 2.04·13-s + (−0.0788 − 1.88i)14-s + (3.94 − 0.665i)16-s − 3.61i·17-s + (1.41 − 0.0591i)18-s + 5.35i·19-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0418i)2-s + 0.577·3-s + (0.996 − 0.0835i)4-s + (0.576 − 0.0241i)6-s − 0.504i·7-s + (0.992 − 0.125i)8-s + 0.333·9-s − 0.887i·11-s + (0.575 − 0.0482i)12-s − 0.566·13-s + (−0.0210 − 0.503i)14-s + (0.986 − 0.166i)16-s − 0.876i·17-s + (0.333 − 0.0139i)18-s + 1.22i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.22440 - 0.550519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.22440 - 0.550519i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0591i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.33iT - 7T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.61iT - 17T^{2} \) |
| 19 | \( 1 - 5.35iT - 19T^{2} \) |
| 23 | \( 1 - 8.59iT - 23T^{2} \) |
| 29 | \( 1 - 5.26iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 + 9.97iT - 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 - 4.75iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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
−10.68282207143527938738732743944, −9.936092604230651304639103313263, −8.835110364653254064812909049438, −7.63962422382198120006857997793, −7.15002834793942196828727274863, −5.88175235268422479899289774330, −5.02809339250815931590734770693, −3.76862831431111336616735592613, −3.10983942703431154316713992371, −1.62156460732819543500761250582,
2.03310905206904008155136504438, 2.86085569548846092714062913626, 4.23634401932990061232852529177, 4.91841396472312172419833378779, 6.18459397964407310273768024465, 7.00615887324707667942640613326, 7.922339506644431173764879170292, 8.899421555619399293029814345359, 9.959243276377854374841850050073, 10.78193691255046807763068438041