L(s) = 1 | + (0.805 − 1.39i)2-s + (1.04 − 1.38i)3-s + (−0.296 − 0.513i)4-s − 5-s + (−1.09 − 2.56i)6-s + (2.48 − 0.911i)7-s + 2.26·8-s + (−0.832 − 2.88i)9-s + (−0.805 + 1.39i)10-s − 2.16·11-s + (−1.02 − 0.124i)12-s + (−2.21 + 3.83i)13-s + (0.729 − 4.19i)14-s + (−1.04 + 1.38i)15-s + (2.41 − 4.18i)16-s + (−0.752 + 1.30i)17-s + ⋯ |
L(s) = 1 | + (0.569 − 0.986i)2-s + (0.601 − 0.799i)3-s + (−0.148 − 0.256i)4-s − 0.447·5-s + (−0.445 − 1.04i)6-s + (0.938 − 0.344i)7-s + 0.800·8-s + (−0.277 − 0.960i)9-s + (−0.254 + 0.441i)10-s − 0.653·11-s + (−0.294 − 0.0358i)12-s + (−0.613 + 1.06i)13-s + (0.194 − 1.12i)14-s + (−0.268 + 0.357i)15-s + (0.604 − 1.04i)16-s + (−0.182 + 0.316i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26469 - 1.72089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26469 - 1.72089i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.04 + 1.38i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.48 + 0.911i)T \) |
good | 2 | \( 1 + (-0.805 + 1.39i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + (2.21 - 3.83i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.752 - 1.30i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.165 - 0.286i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.87 - 8.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.09 + 1.90i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.44 - 5.96i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.16 - 10.6i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.908 + 1.57i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.32 + 2.29i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.78 + 6.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.43 + 4.21i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.61 + 11.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + (5.31 - 9.20i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.55 - 4.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.99 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.99 + 5.18i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.11 - 5.39i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.60034327826753717921880877388, −10.83966860686863450023244717732, −9.666063971934519105125483348307, −8.282730816424490119434370915286, −7.70711339111322111619783056865, −6.71426726863979420757057131468, −4.90688737419908448730301636821, −3.95643201972230935672976197738, −2.66677954901861024906776472195, −1.57633362447262430553327146634,
2.47732149279625714994076850671, 4.07922449792286357847527336913, 5.03809634153918541291176160031, 5.66558451506474659112401761295, 7.41054102344552324379462010244, 7.892184497429209798019060455357, 8.827992645744516064017838970041, 10.20813668202660258211698176739, 10.80749415049506222593904686951, 11.95354726440844574656852832642