L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.16 + 1.16i)3-s − 1.00i·4-s − 1.64·6-s + (2.19 + 2.19i)7-s + (0.707 + 0.707i)8-s − 0.304i·9-s + 4.06·11-s + (1.16 − 1.16i)12-s + (−0.238 − 0.238i)13-s − 3.10·14-s − 1.00·16-s + (2.63 + 2.63i)17-s + (0.215 + 0.215i)18-s + (0.420 − 4.33i)19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.670 + 0.670i)3-s − 0.500i·4-s − 0.670·6-s + (0.828 + 0.828i)7-s + (0.250 + 0.250i)8-s − 0.101i·9-s + 1.22·11-s + (0.335 − 0.335i)12-s + (−0.0660 − 0.0660i)13-s − 0.828·14-s − 0.250·16-s + (0.638 + 0.638i)17-s + (0.0508 + 0.0508i)18-s + (0.0965 − 0.995i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43397 + 1.13955i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43397 + 1.13955i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.420 + 4.33i)T \) |
good | 3 | \( 1 + (-1.16 - 1.16i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.19 - 2.19i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + (0.238 + 0.238i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.63 - 2.63i)T + 17iT^{2} \) |
| 23 | \( 1 + (-2.60 + 2.60i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.78T + 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-3.07 + 3.07i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.82iT - 41T^{2} \) |
| 43 | \( 1 + (4.02 - 4.02i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.81 + 1.81i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.19 - 6.19i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.737T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (8.99 - 8.99i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (-3.94 + 3.94i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 + (10.2 - 10.2i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + (8.40 - 8.40i)T - 97iT^{2} \) |
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
−9.870748652143544931415140372767, −9.167573954451209417573243735447, −8.740870116819556654024680777974, −7.967065683100021504702581791887, −6.88328877967394003680598329306, −5.98538879862023437211711407143, −4.95969239863586394327627694374, −4.02785848327757733460140636504, −2.82566952661212471173804099243, −1.40755997011801257917257541089,
1.18157036726267677656068190836, 1.94890353757646712856702142414, 3.34569709850825012855829699939, 4.21518667489877874842726101312, 5.48094717355740746803660299457, 6.93438483571786385361353646288, 7.50178257372088269013336240301, 8.136259144493318461919532387017, 9.008795137643638297021148576220, 9.780585504549805353949233761797