| L(s) = 1 | + (−0.239 − 0.153i)2-s + (−2.47 − 0.727i)3-s + (−0.797 − 1.74i)4-s + (1.21 − 1.40i)5-s + (0.481 + 0.555i)6-s + (−1.14 − 0.737i)7-s + (−0.158 + 1.10i)8-s + (3.09 + 1.98i)9-s + (−0.508 + 0.149i)10-s + (1.59 − 1.84i)11-s + (0.705 + 4.90i)12-s + (0.666 + 4.63i)13-s + (0.161 + 0.353i)14-s + (−4.04 + 2.60i)15-s + (−2.30 + 2.66i)16-s + (2.99 − 6.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.169 − 0.108i)2-s + (−1.43 − 0.420i)3-s + (−0.398 − 0.872i)4-s + (0.545 − 0.629i)5-s + (0.196 + 0.226i)6-s + (−0.433 − 0.278i)7-s + (−0.0561 + 0.390i)8-s + (1.03 + 0.662i)9-s + (−0.160 + 0.0472i)10-s + (0.481 − 0.555i)11-s + (0.203 + 1.41i)12-s + (0.184 + 1.28i)13-s + (0.0431 + 0.0944i)14-s + (−1.04 + 0.671i)15-s + (−0.576 + 0.665i)16-s + (0.726 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.314111 - 0.422379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.314111 - 0.422379i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-7.91 - 2.09i)T \) |
| good | 2 | \( 1 + (0.239 + 0.153i)T + (0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (2.47 + 0.727i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-1.21 + 1.40i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (1.14 + 0.737i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 1.84i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.666 - 4.63i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.99 + 6.56i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-4.79 + 3.08i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (5.17 + 1.52i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 + (-0.185 + 1.28i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + 4.48T + 37T^{2} \) |
| 41 | \( 1 + (-1.16 + 2.55i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.94 - 10.8i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-6.10 - 1.79i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (-0.682 - 1.49i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.41 - 9.85i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.26 - 2.61i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (0.618 + 1.35i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (2.13 + 2.46i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.35 - 9.43i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-9.49 + 10.9i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.09 + 1.20i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + 7.56T + 97T^{2} \) |
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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
−14.09126767743642698219433408116, −13.52082553205531804092306338519, −12.00690801533114403055232835515, −11.31766377946172628748393863807, −9.937607933903271082804651669314, −9.109343492410530694110495434254, −6.87807753430090122158673654198, −5.82720502939292000045132158003, −4.83326736093166722115780372971, −1.02168903398244016322867281292,
3.64124888433491507815836276525, 5.47689129268290084321979165290, 6.50686735820698861983259654861, 8.074391226696433617267486880304, 9.861338918502221573878434883130, 10.45880105628394997792340188942, 12.04962925669544894748396857670, 12.52560645541792283202088454530, 14.01383275201992735705825739132, 15.48729478383121400528982989763
