| L(s) = 1 | + (0.288 + 0.344i)2-s + (0.669 − 1.83i)3-s + (0.312 − 1.77i)4-s + (−0.869 + 2.05i)5-s + (0.826 − 0.300i)6-s + (−1.78 − 1.03i)7-s + (1.47 − 0.853i)8-s + (−0.635 − 0.533i)9-s + (−0.960 + 0.295i)10-s + (1.15 + 1.99i)11-s + (−3.04 − 1.75i)12-s + (1.50 + 4.13i)13-s + (−0.160 − 0.911i)14-s + (3.20 + 2.97i)15-s + (−2.65 − 0.967i)16-s + (3.51 + 4.19i)17-s + ⋯ |
| L(s) = 1 | + (0.204 + 0.243i)2-s + (0.386 − 1.06i)3-s + (0.156 − 0.885i)4-s + (−0.389 + 0.921i)5-s + (0.337 − 0.122i)6-s + (−0.674 − 0.389i)7-s + (0.522 − 0.301i)8-s + (−0.211 − 0.177i)9-s + (−0.303 + 0.0934i)10-s + (0.347 + 0.602i)11-s + (−0.879 − 0.507i)12-s + (0.417 + 1.14i)13-s + (−0.0429 − 0.243i)14-s + (0.827 + 0.768i)15-s + (−0.664 − 0.241i)16-s + (0.853 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09949 - 0.388736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09949 - 0.388736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.869 - 2.05i)T \) |
| 19 | \( 1 + (3.07 + 3.09i)T \) |
| good | 2 | \( 1 + (-0.288 - 0.344i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.669 + 1.83i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (1.78 + 1.03i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.15 - 1.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.50 - 4.13i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.51 - 4.19i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.72 + 1.01i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.21 + 3.53i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.378 - 0.656i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.22iT - 37T^{2} \) |
| 41 | \( 1 + (-6.14 - 2.23i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.11 - 0.549i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.08 + 4.87i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (3.79 + 0.668i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (1.95 - 1.63i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.72 + 9.78i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.09 + 1.30i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.71 - 9.70i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.49 + 9.60i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.26 - 0.824i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.11 - 3.53i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.19 - 0.798i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (4.38 + 5.22i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.98240452379519152094534167592, −13.08873448536019941613359258133, −11.85942400790458942112819061013, −10.63564109825493634487613648252, −9.669595526893279856360387614120, −7.928907884541107563264450267216, −6.77810396902772743092921886629, −6.35151485957982930226321662838, −4.06766049917523334756734696768, −1.97019657274504792693066633238,
3.24677354322292019858390107872, 4.08627077458481092970152173150, 5.64923847963979429866344007346, 7.72304875876020930552050613993, 8.718798541232039813438085594185, 9.626737836508702707743817270404, 10.94328820363811789230918867671, 12.23478743755380200245807876240, 12.78343559879836214070881469644, 14.05541920159585529887989394852
