| L(s) = 1 | + (−0.358 − 0.933i)2-s + (−1.20 − 1.85i)3-s + (−0.743 + 0.669i)4-s + (1.76 + 1.36i)5-s + (−1.29 + 1.78i)6-s + (−2.10 − 1.60i)7-s + (0.891 + 0.453i)8-s + (−0.764 + 1.71i)9-s + (0.641 − 2.14i)10-s + (1.56 − 2.92i)11-s + (2.13 + 0.571i)12-s + (−2.41 − 0.381i)13-s + (−0.745 + 2.53i)14-s + (0.402 − 4.92i)15-s + (0.104 − 0.994i)16-s + (−0.578 + 1.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.660i)2-s + (−0.694 − 1.06i)3-s + (−0.371 + 0.334i)4-s + (0.791 + 0.611i)5-s + (−0.530 + 0.729i)6-s + (−0.794 − 0.606i)7-s + (0.315 + 0.160i)8-s + (−0.254 + 0.572i)9-s + (0.202 − 0.677i)10-s + (0.470 − 0.882i)11-s + (0.615 + 0.165i)12-s + (−0.668 − 0.105i)13-s + (−0.199 + 0.678i)14-s + (0.104 − 1.27i)15-s + (0.0261 − 0.248i)16-s + (−0.140 + 0.365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.183182 + 0.447728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183182 + 0.447728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 + 0.933i)T \) |
| 5 | \( 1 + (-1.76 - 1.36i)T \) |
| 7 | \( 1 + (2.10 + 1.60i)T \) |
| 11 | \( 1 + (-1.56 + 2.92i)T \) |
| good | 3 | \( 1 + (1.20 + 1.85i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (2.41 + 0.381i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.578 - 1.50i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-2.01 + 2.24i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (8.98 + 2.40i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (6.48 + 2.10i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.72 + 0.601i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (3.42 + 2.22i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (1.53 - 0.498i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.77 - 4.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.165 + 3.15i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-9.74 - 7.89i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (0.346 + 0.384i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (6.65 + 0.699i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (3.51 - 0.942i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.855 - 0.621i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.531 + 10.1i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (2.67 - 5.99i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.546 - 3.45i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (5.21 + 9.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.833 + 5.26i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−9.967501960991343405381233795840, −9.199564044305234422628579446670, −7.928125658530382840923387408842, −7.04634408265897657629487083517, −6.32102358579437750143006102364, −5.66315128453489214421052316004, −4.01131183446279479127066518458, −2.85693123701797620351553137184, −1.66913120362514125178067689194, −0.28084117822077556589440418452,
1.99430957506096124649244652416, 3.83972777200680828663069077910, 4.85591298655948373273419815528, 5.52603769221016734354605576377, 6.22763421931893531551842213523, 7.26291922698334790510348147546, 8.510838625078209305174476697130, 9.508779380353490415430316349654, 9.786156653322660402009952633207, 10.31871136247107615832306805953
