L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 0.732i·7-s − i·8-s − 9-s − 5.19·11-s − i·12-s − 1.26i·13-s + 0.732·14-s + 16-s + 0.732i·17-s − i·18-s − 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.276i·7-s − 0.353i·8-s − 0.333·9-s − 1.56·11-s − 0.288i·12-s − 0.351i·13-s + 0.195·14-s + 0.250·16-s + 0.177i·17-s − 0.235i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.210520301\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.210520301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26iT - 13T^{2} \) |
| 17 | \( 1 - 0.732iT - 17T^{2} \) |
| 23 | \( 1 + 1.53iT - 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 4.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.26iT - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 0.196T + 71T^{2} \) |
| 73 | \( 1 + 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 - 0.660iT - 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 7.12iT - 97T^{2} \) |
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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
−8.486300202624751415243784382714, −8.251998591654184915508128480728, −7.36984445443009741892161292921, −6.56639758333385282195374054381, −5.64373291278021850498905695919, −5.06418141201740278329540369789, −4.30564504719904978926365699205, −3.31750153149361172839494710818, −2.35384889790936553554666113307, −0.51228004905576739087550179187,
0.857521815949493778224528146501, 2.22644081173563451039387764782, 2.69281341699373771023605708743, 3.82324098926574637040802474343, 4.87795904039429111512840537185, 5.53137622768149060630615426801, 6.40637654726677250701845981002, 7.40894021389042160244157939043, 8.008513065768709079574155694396, 8.717479274591095405505704382379