L(s) = 1 | + (−0.380 − 2.40i)2-s + (0.532 − 0.271i)3-s + (−3.72 + 1.21i)4-s + (−0.622 − 2.14i)5-s + (−0.855 − 1.17i)6-s + (−1.17 + 2.30i)7-s + (2.11 + 4.15i)8-s + (−1.55 + 2.13i)9-s + (−4.92 + 2.31i)10-s + (−1.65 + 1.65i)12-s + (−2.77 + 0.439i)13-s + (5.97 + 1.94i)14-s + (−0.914 − 0.975i)15-s + (2.84 − 2.06i)16-s + (0.932 + 0.147i)17-s + (5.72 + 2.91i)18-s + ⋯ |
L(s) = 1 | + (−0.269 − 1.69i)2-s + (0.307 − 0.156i)3-s + (−1.86 + 0.605i)4-s + (−0.278 − 0.960i)5-s + (−0.349 − 0.480i)6-s + (−0.443 + 0.869i)7-s + (0.748 + 1.46i)8-s + (−0.517 + 0.712i)9-s + (−1.55 + 0.731i)10-s + (−0.478 + 0.478i)12-s + (−0.769 + 0.121i)13-s + (1.59 + 0.518i)14-s + (−0.236 − 0.251i)15-s + (0.710 − 0.516i)16-s + (0.226 + 0.0358i)17-s + (1.34 + 0.687i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117384 + 0.0424897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117384 + 0.0424897i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.622 + 2.14i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.380 + 2.40i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.532 + 0.271i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.17 - 2.30i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (2.77 - 0.439i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.932 - 0.147i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (1.28 - 3.94i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.104 - 0.104i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.14 + 6.60i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.33 + 5.32i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.77 - 3.44i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (3.27 + 1.06i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (3.91 - 3.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.479 - 0.942i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.644 + 4.07i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (6.16 - 2.00i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.59 + 7.70i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \) |
| 71 | \( 1 + (1.02 - 0.744i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.57 + 0.804i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-3.51 - 2.55i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.29 - 8.18i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 4.23iT - 89T^{2} \) |
| 97 | \( 1 + (14.2 - 2.25i)T + (92.2 - 29.9i)T^{2} \) |
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
−10.89647236949693466430519122857, −9.678469126290664279621013720584, −9.387045516205761565076047977895, −8.369021027667460028518534590602, −7.80544038615367905914521730540, −5.89258547596888755623808064179, −4.88152735993752257073320637222, −3.81148174316419752179760104704, −2.64247440105600960296766763517, −1.77584913056679898184491117561,
0.07009834972532837411703049205, 3.01819074345638089159645782917, 4.07222173164385900878029545287, 5.31490617706623701050427199715, 6.38624809187010808008595162654, 7.08013121567981357898194362189, 7.53775551697327506017767603549, 8.684741582229723776079494599062, 9.393240638982566612823765203454, 10.26621141409424501531719687588