| L(s) = 1 | + (−0.939 + 0.581i)2-s + (0.273 + 0.961i)3-s + (−0.347 + 0.697i)4-s + (−2.57 + 0.998i)5-s + (−0.816 − 0.744i)6-s + (0.179 − 1.93i)7-s + (−0.283 − 3.05i)8-s + (−0.850 + 0.526i)9-s + (1.84 − 2.43i)10-s + (−4.16 + 2.58i)11-s + (−0.766 − 0.143i)12-s + (0.198 − 2.14i)13-s + (0.957 + 1.92i)14-s + (−1.66 − 2.20i)15-s + (1.10 + 1.46i)16-s + (4.14 − 3.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.411i)2-s + (0.157 + 0.555i)3-s + (−0.173 + 0.348i)4-s + (−1.15 + 0.446i)5-s + (−0.333 − 0.303i)6-s + (0.0677 − 0.731i)7-s + (−0.100 − 1.08i)8-s + (−0.283 + 0.175i)9-s + (0.581 − 0.770i)10-s + (−1.25 + 0.778i)11-s + (−0.221 − 0.0413i)12-s + (0.0551 − 0.595i)13-s + (0.255 + 0.513i)14-s + (−0.430 − 0.569i)15-s + (0.276 + 0.365i)16-s + (1.00 − 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0152652 - 0.0227420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0152652 - 0.0227420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.273 - 0.961i)T \) |
| 103 | \( 1 + (-6.56 - 7.73i)T \) |
| good | 2 | \( 1 + (0.939 - 0.581i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (2.57 - 0.998i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.179 + 1.93i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (4.16 - 2.58i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.198 + 2.14i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (-4.14 + 3.77i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.0521 - 0.183i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (1.63 + 1.00i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (7.91 - 3.06i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (1.90 + 2.52i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (5.60 + 1.04i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (3.88 + 1.50i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (-0.775 + 0.144i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + (1.91 - 6.73i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.792 + 8.54i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (6.72 - 6.13i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.429 - 4.63i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-11.7 - 4.55i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-3.26 - 1.26i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (14.5 - 5.62i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (1.16 - 12.5i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (0.854 + 1.71i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (3.16 + 2.88i)T + (8.95 + 96.5i)T^{2} \) |
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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
−12.30457847013780477066060078324, −11.17428629076156293644457458353, −10.29248045168812752724129832990, −9.589818329107984628519807618573, −8.285568874669997665901746001366, −7.58916976570918255526502765303, −7.17523071583897994148942581935, −5.23694266323506682795621051881, −4.00561096106914972111704245789, −3.15061114446551785936909607007,
0.02380434454613079424855392540, 1.82877306845276051503542072744, 3.42237232514480367844207705062, 5.07661927676701378846273682401, 5.98078728604293485186599064608, 7.70499215192312146319527004684, 8.253343728838451447666465600574, 8.960312956107617648457615944193, 10.12834462832330873685794346526, 11.16174940513530559880228931298
