L(s) = 1 | + (0.474 + 0.127i)2-s + (0.708 − 1.58i)3-s + (−1.52 − 0.879i)4-s + (−2.23 − 0.0580i)5-s + (0.536 − 0.659i)6-s + (−1.30 − 1.30i)8-s + (−1.99 − 2.24i)9-s + (−1.05 − 0.311i)10-s + (2.31 + 1.33i)11-s + (−2.46 + 1.78i)12-s + (−2.14 + 2.14i)13-s + (−1.67 + 3.49i)15-s + (1.30 + 2.26i)16-s + (−1.19 − 4.46i)17-s + (−0.661 − 1.31i)18-s + (−4.54 + 2.62i)19-s + ⋯ |
L(s) = 1 | + (0.335 + 0.0898i)2-s + (0.409 − 0.912i)3-s + (−0.761 − 0.439i)4-s + (−0.999 − 0.0259i)5-s + (0.219 − 0.269i)6-s + (−0.461 − 0.461i)8-s + (−0.665 − 0.746i)9-s + (−0.332 − 0.0984i)10-s + (0.697 + 0.402i)11-s + (−0.712 + 0.515i)12-s + (−0.596 + 0.596i)13-s + (−0.432 + 0.901i)15-s + (0.326 + 0.565i)16-s + (−0.290 − 1.08i)17-s + (−0.155 − 0.310i)18-s + (−1.04 + 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0711015 + 0.203897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0711015 + 0.203897i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.708 + 1.58i)T \) |
| 5 | \( 1 + (2.23 + 0.0580i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.474 - 0.127i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (-2.31 - 1.33i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.14 - 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.19 + 4.46i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.932 - 3.48i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.784 - 2.92i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \) |
| 47 | \( 1 + (10.4 + 2.80i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.05 - 1.62i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.0797 + 0.138i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 + 1.98i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (1.52 + 5.68i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.37 + 1.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.03 + 4.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.97 - 3.42i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \) |
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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
−9.536548783522818726879480994732, −8.996297047159592005596791617210, −8.121014361633173693781721737431, −7.14987312062344512124872732923, −6.54077218902372834585875457526, −5.25689054573318403276658250781, −4.24676998679283941579963824861, −3.38832420351982770024336276161, −1.75126930186022261815463491893, −0.094761787576987821527042633521,
2.70341084412051124621407716994, 3.75386910330076629268291497410, 4.26515325848843354577050457594, 5.13533154709833785086291683403, 6.41739986591686545624801691988, 7.890579895201887210209427246077, 8.333967778794565978880684855549, 9.087014956356051299041552762884, 9.962860845484752508266534849496, 11.03669513467290968262692554703