L(s) = 1 | + (0.435 − 0.221i)2-s + (1.12 + 0.178i)3-s + (−1.03 + 1.42i)4-s + (−1.14 − 1.92i)5-s + (0.529 − 0.171i)6-s + (−0.456 − 2.88i)7-s + (−0.287 + 1.81i)8-s + (−1.61 − 0.525i)9-s + (−0.922 − 0.583i)10-s + (−1.41 + 1.41i)12-s + (−1.32 − 2.60i)13-s + (−0.837 − 1.15i)14-s + (−0.940 − 2.36i)15-s + (−0.811 − 2.49i)16-s + (2.39 − 4.69i)17-s + (−0.820 + 0.130i)18-s + ⋯ |
L(s) = 1 | + (0.307 − 0.156i)2-s + (0.649 + 0.102i)3-s + (−0.517 + 0.712i)4-s + (−0.510 − 0.860i)5-s + (0.216 − 0.0701i)6-s + (−0.172 − 1.08i)7-s + (−0.101 + 0.641i)8-s + (−0.539 − 0.175i)9-s + (−0.291 − 0.184i)10-s + (−0.409 + 0.409i)12-s + (−0.368 − 0.723i)13-s + (−0.223 − 0.307i)14-s + (−0.242 − 0.611i)15-s + (−0.202 − 0.624i)16-s + (0.580 − 1.13i)17-s + (−0.193 + 0.0306i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.724508 - 0.968843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.724508 - 0.968843i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.14 + 1.92i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.435 + 0.221i)T + (1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-1.12 - 0.178i)T + (2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (0.456 + 2.88i)T + (-6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (1.32 + 2.60i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.39 + 4.69i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 2.40i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.12 + 2.12i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2.13 - 1.55i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.08 - 6.41i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.14 - 0.181i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.08 - 2.87i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.07 + 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.575 + 3.63i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.31 + 0.670i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (0.943 - 1.29i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.59 - 2.14i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.887 - 2.73i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (14.6 - 2.31i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.71 + 14.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.95 - 3.03i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (4.57 + 8.97i)T + (-57.0 + 78.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30050483949521598248731396062, −9.282941810230008648464492700923, −8.686094808217239982701545493142, −7.76635365941391301024431437066, −7.26038689916755113217488971216, −5.42650095693347120587483795842, −4.61424088550816450178166348201, −3.60963948810782938858658343805, −2.89914625366935299535103887825, −0.56175705524728578125601297422,
2.04599684957305120377036694940, 3.23029290099596222859511299106, 4.23146843489574890506811709772, 5.68712426793760390315111548302, 6.10947407662814450110182783133, 7.46252828232219009930301356997, 8.277307037788234015178184984037, 9.221747525401927638480462672334, 9.863902465307796986419360934128, 10.89915481170493338485547049983