L(s) = 1 | + (−0.740 + 1.66i)2-s + (2.39 − 2.15i)3-s + (−0.878 − 0.975i)4-s + (1.81 + 1.30i)5-s + (1.81 + 5.57i)6-s + (−1.22 − 2.34i)7-s + (−1.18 + 0.386i)8-s + (0.771 − 7.34i)9-s + (−3.51 + 2.05i)10-s + (0.271 + 2.58i)11-s + (−4.20 − 0.442i)12-s + (0.191 − 0.263i)13-s + (4.80 − 0.296i)14-s + (7.16 − 0.784i)15-s + (0.512 − 4.87i)16-s + (−0.205 + 0.968i)17-s + ⋯ |
L(s) = 1 | + (−0.523 + 1.17i)2-s + (1.38 − 1.24i)3-s + (−0.439 − 0.487i)4-s + (0.811 + 0.584i)5-s + (0.739 + 2.27i)6-s + (−0.462 − 0.886i)7-s + (−0.420 + 0.136i)8-s + (0.257 − 2.44i)9-s + (−1.11 + 0.648i)10-s + (0.0818 + 0.778i)11-s + (−1.21 − 0.127i)12-s + (0.0530 − 0.0730i)13-s + (1.28 − 0.0791i)14-s + (1.84 − 0.202i)15-s + (0.128 − 1.21i)16-s + (−0.0499 + 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34408 + 0.380376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34408 + 0.380376i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.81 - 1.30i)T \) |
| 7 | \( 1 + (1.22 + 2.34i)T \) |
good | 2 | \( 1 + (0.740 - 1.66i)T + (-1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-2.39 + 2.15i)T + (0.313 - 2.98i)T^{2} \) |
| 11 | \( 1 + (-0.271 - 2.58i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.191 + 0.263i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.205 - 0.968i)T + (-15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (5.14 - 5.71i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.166 - 0.373i)T + (-15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (1.03 - 3.19i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.917 + 0.194i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (2.04 + 0.215i)T + (36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-4.92 - 3.57i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.80iT - 43T^{2} \) |
| 47 | \( 1 + (2.09 + 9.84i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.94 + 3.55i)T + (5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (3.91 - 1.74i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (6.64 + 2.95i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (0.0193 - 0.0910i)T + (-61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-1.38 + 4.26i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.43 + 0.360i)T + (71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (4.51 - 0.959i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.90 - 0.942i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-8.40 - 3.74i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-6.26 - 2.03i)T + (78.4 + 57.0i)T^{2} \) |
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
−13.10414313080490211949899195082, −12.30100866232252462134527227064, −10.32804715594951496919512484552, −9.394424724937759732023839838901, −8.439549910773319440309625426537, −7.46994118097924463078495540723, −6.87185230018358184025613723715, −6.10474533300010561213367112898, −3.50047793304922209359485990452, −1.99326246047434164919480408635,
2.25383416062107155028304906171, 3.04528565498362626562354022295, 4.50931401005146139279015540387, 6.01919495763805826848805240649, 8.381797196905148840400401095854, 9.090453404235850900339293064357, 9.414082956781634737913523774921, 10.40908069777706730220401823152, 11.25885109050440414575924187332, 12.70897286699567924792791989890