| L(s) = 1 | + (−0.288 − 0.344i)2-s + (−0.669 + 1.83i)3-s + (0.312 − 1.77i)4-s + (1.52 + 1.63i)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 + (0.124 − 0.997i)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 + (−4.03 + 1.70i)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 + (0.680 + 0.732i)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 + (0.0393 − 0.315i)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 + (−1.04 + 0.439i)15-s + (−0.664 − 0.241i)16-s + (−0.853 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.909377 + 0.217020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.909377 + 0.217020i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.52 - 1.63i)T \) |
| 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
−14.49892229132148805078833687368, −13.07875659849749680131415725990, −11.26505089200642448931528340143, −10.91557236828665859066442948020, −9.837197439990403830836418950871, −9.180871396852235333528967540380, −7.10511862558977805219096362613, −5.64508029306244709910060937204, −4.76992599671801651916254911748, −2.41492624446593803996216237237,
1.76789054412285019256508380258, 4.30042895607646597345792661514, 6.16392489881092556921393996662, 7.06096942556759586377758240460, 8.287790680627514533139960615081, 9.156676480687663832929975779938, 10.99685637640413241067398185256, 12.04041945863198182515804637723, 12.85012321547319703239826268322, 13.56582101803992074667453892916
