| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 13-s + 15-s + 16-s + 5·17-s − 18-s + 7·19-s − 20-s − 4·22-s − 8·23-s + 24-s + 25-s + 26-s − 27-s + 2·29-s − 30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.20473689597745, −15.70434738818694, −14.93771573542287, −14.37488778682874, −14.02684033339040, −13.16389010527193, −12.38513898587664, −11.94613101538855, −11.58040934263639, −11.24586968910449, −10.15532978084654, −9.850627658397516, −9.575835960961613, −8.471254242618264, −8.213620673081798, −7.391246931082544, −6.975223568421440, −6.326231338671939, −5.588286859238234, −5.111616261375831, −4.044733541422618, −3.612900523575043, −2.751707314837597, −1.592392976757588, −1.071578594894199, 0,
1.071578594894199, 1.592392976757588, 2.751707314837597, 3.612900523575043, 4.044733541422618, 5.111616261375831, 5.588286859238234, 6.326231338671939, 6.975223568421440, 7.391246931082544, 8.213620673081798, 8.471254242618264, 9.575835960961613, 9.850627658397516, 10.15532978084654, 11.24586968910449, 11.58040934263639, 11.94613101538855, 12.38513898587664, 13.16389010527193, 14.02684033339040, 14.37488778682874, 14.93771573542287, 15.70434738818694, 16.20473689597745
