L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (1.42 − 1.72i)5-s − 1.00·6-s + (3.40 + 3.40i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−0.215 − 2.22i)10-s + 3.94·11-s + (−0.707 + 0.707i)12-s + (4.03 + 4.03i)13-s + 4.81·14-s + (−2.22 + 0.215i)15-s − 1.00·16-s + (−3.90 − 3.90i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.635 − 0.772i)5-s − 0.408·6-s + (1.28 + 1.28i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.0682 − 0.703i)10-s + 1.18·11-s + (−0.204 + 0.204i)12-s + (1.11 + 1.11i)13-s + 1.28·14-s + (−0.574 + 0.0557i)15-s − 0.250·16-s + (−0.947 − 0.947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86477 - 1.10060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86477 - 1.10060i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.42 + 1.72i)T \) |
| 19 | \( 1 + (3.49 - 2.60i)T \) |
good | 7 | \( 1 + (-3.40 - 3.40i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 + (-4.03 - 4.03i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.90 + 3.90i)T + 17iT^{2} \) |
| 23 | \( 1 + (-0.537 + 0.537i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.28T + 29T^{2} \) |
| 31 | \( 1 + 7.00iT - 31T^{2} \) |
| 37 | \( 1 + (-1.05 + 1.05i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.03iT - 41T^{2} \) |
| 43 | \( 1 + (5.70 - 5.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.04 + 2.04i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.39 + 4.39i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.32T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 + (-7.35 + 7.35i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.62iT - 71T^{2} \) |
| 73 | \( 1 + (-4.47 + 4.47i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.991T + 79T^{2} \) |
| 83 | \( 1 + (-6.64 + 6.64i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.09T + 89T^{2} \) |
| 97 | \( 1 + (7.11 - 7.11i)T - 97iT^{2} \) |
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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
−11.23568939518729060333265734370, −9.504346779355957152005859915522, −8.981757640352600795179316231211, −8.185553441125115500751702755339, −6.52465316918755560089009272632, −5.94196354496356848322077489291, −4.93953042963332345556925673974, −4.16655660181977281474248914850, −2.11436836763297951496611674161, −1.54337266397143508784882773733,
1.57866209089037515794718852425, 3.54375657100399274393155139920, 4.26413591631868548760469778835, 5.37332260372996531228561007773, 6.39209514576320259694748170907, 6.99391361086851408627643491467, 8.141245968335343489890978952219, 9.012982004789773966182643163831, 10.40625619313041246956532011439, 10.89884778185843634027116947673