L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.966 − 1.43i)3-s − 1.00i·4-s + (1.74 + 1.39i)5-s + (−1.69 − 0.333i)6-s + (0.371 + 0.371i)7-s + (−0.707 − 0.707i)8-s + (−1.13 + 2.77i)9-s + (2.22 − 0.248i)10-s − 4.73i·11-s + (−1.43 + 0.966i)12-s + (3.07 − 3.07i)13-s + 0.524·14-s + (0.318 − 3.85i)15-s − 1.00·16-s + (2.77 − 2.77i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.557 − 0.829i)3-s − 0.500i·4-s + (0.781 + 0.624i)5-s + (−0.693 − 0.136i)6-s + (0.140 + 0.140i)7-s + (−0.250 − 0.250i)8-s + (−0.377 + 0.925i)9-s + (0.702 − 0.0785i)10-s − 1.42i·11-s + (−0.414 + 0.278i)12-s + (0.852 − 0.852i)13-s + 0.140·14-s + (0.0822 − 0.996i)15-s − 0.250·16-s + (0.672 − 0.672i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04117 - 1.43310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04117 - 1.43310i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.966 + 1.43i)T \) |
| 5 | \( 1 + (-1.74 - 1.39i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-0.371 - 0.371i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 13 | \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.77 + 2.77i)T - 17iT^{2} \) |
| 23 | \( 1 + (5.62 + 5.62i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 + (-2.20 - 2.20i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.41iT - 41T^{2} \) |
| 43 | \( 1 + (6.80 - 6.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.86 + 5.86i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.65 - 6.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.00T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 + (-4.01 - 4.01i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.67iT - 71T^{2} \) |
| 73 | \( 1 + (-7.51 + 7.51i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.93iT - 79T^{2} \) |
| 83 | \( 1 + (-7.74 - 7.74i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 + (-5.01 - 5.01i)T + 97iT^{2} \) |
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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
−10.67508772901676642783482159417, −10.00453560381271717581456417030, −8.617658274343096336939418847267, −7.77011630248276294622309495551, −6.37314180326001616731161327984, −5.99726971808486604823099008257, −5.15299021855709724460242832476, −3.41693362795432845147513524329, −2.44128553240257785877686219333, −0.999469044093584078282625609016,
1.81358752487021528064355574797, 3.79266051915517596417885575156, 4.50084245500302125035838687749, 5.49281661100965905128015535680, 6.13878810458231858359374664309, 7.21741131173244501952030923746, 8.447885213174820063288474248919, 9.419954582217071205779812687941, 9.952239758021891741044311624154, 10.99161794352078887738523059204