L(s) = 1 | + (0.286 − 0.165i)2-s + (2.33 − 1.34i)3-s + (−0.945 + 1.63i)4-s + (−2.12 + 0.702i)5-s + (0.445 − 0.771i)6-s + (−2.90 − 1.67i)7-s + 1.28i·8-s + (2.12 − 3.67i)9-s + (−0.492 + 0.552i)10-s + (1.62 + 2.81i)11-s + 5.08i·12-s + (1.21 − 3.39i)13-s − 1.10·14-s + (−4.00 + 4.49i)15-s + (−1.67 − 2.90i)16-s + (1.68 + 0.974i)17-s + ⋯ |
L(s) = 1 | + (0.202 − 0.116i)2-s + (1.34 − 0.777i)3-s + (−0.472 + 0.818i)4-s + (−0.949 + 0.314i)5-s + (0.181 − 0.314i)6-s + (−1.09 − 0.633i)7-s + 0.455i·8-s + (0.707 − 1.22i)9-s + (−0.155 + 0.174i)10-s + (0.489 + 0.847i)11-s + 1.46i·12-s + (0.337 − 0.941i)13-s − 0.296·14-s + (−1.03 + 1.16i)15-s + (−0.419 − 0.726i)16-s + (0.409 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07821 - 0.151959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07821 - 0.151959i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.12 - 0.702i)T \) |
| 13 | \( 1 + (-1.21 + 3.39i)T \) |
good | 2 | \( 1 + (-0.286 + 0.165i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.90 + 1.67i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.62 - 2.81i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.68 - 0.974i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.622 - 1.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.33 - 1.34i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + (-1.68 + 0.974i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.39 + 2.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.56 + 4.36i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.86iT - 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 + (1.26 - 2.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.47 - 2.00i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.62 - 4.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.46iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 8.61iT - 83T^{2} \) |
| 89 | \( 1 + (-5.15 - 8.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.56 - 2.63i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.61838851505959220754356372445, −13.61045616296101317179177789938, −12.79674749382801465609686808840, −12.05894959399723403687798905906, −10.11059719889506248818840211410, −8.707145576744010728300286398628, −7.77275005599585380531522388616, −6.94925000619478405758208279094, −3.92706211960688257891627126189, −3.08511671997933752977749102447,
3.32363031146889635095663936151, 4.46893513135101342848555581271, 6.35000686641745691679965300630, 8.381738832557396449331456478685, 9.134421742788711533880306226492, 9.965789849738433344693805745657, 11.60313763980110755872533933725, 13.12690427521751775964214235088, 14.08871505668943704235109417102, 14.93420594240587105921005196178