| L(s) = 1 | + (−0.186 − 0.982i)2-s + (0.932 − 0.0218i)3-s + (−0.930 + 0.366i)4-s + (0.272 − 1.03i)5-s + (−0.195 − 0.912i)6-s + (1.94 + 4.33i)7-s + (0.533 + 0.845i)8-s + (−2.12 + 0.0998i)9-s + (−1.06 − 0.0752i)10-s + (3.06 + 3.44i)11-s + (−0.860 + 0.362i)12-s + (−0.431 + 6.12i)13-s + (3.89 − 2.72i)14-s + (0.232 − 0.971i)15-s + (0.731 − 0.681i)16-s + (−3.31 − 2.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.131 − 0.694i)2-s + (0.538 − 0.0126i)3-s + (−0.465 + 0.183i)4-s + (0.122 − 0.462i)5-s + (−0.0797 − 0.372i)6-s + (0.735 + 1.63i)7-s + (0.188 + 0.299i)8-s + (−0.708 + 0.0332i)9-s + (−0.337 − 0.0237i)10-s + (0.922 + 1.03i)11-s + (−0.248 + 0.104i)12-s + (−0.119 + 1.69i)13-s + (1.04 − 0.727i)14-s + (0.0599 − 0.250i)15-s + (0.182 − 0.170i)16-s + (−0.802 − 0.713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55797 + 0.181133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55797 + 0.181133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.186 + 0.982i)T \) |
| 269 | \( 1 + (-16.0 + 3.51i)T \) |
| good | 3 | \( 1 + (-0.932 + 0.0218i)T + (2.99 - 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.272 + 1.03i)T + (-4.34 - 2.46i)T^{2} \) |
| 7 | \( 1 + (-1.94 - 4.33i)T + (-4.65 + 5.23i)T^{2} \) |
| 11 | \( 1 + (-3.06 - 3.44i)T + (-1.28 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.431 - 6.12i)T + (-12.8 - 1.82i)T^{2} \) |
| 17 | \( 1 + (3.31 + 2.94i)T + (1.98 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 0.831i)T + (7.37 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.0343 + 1.46i)T + (-22.9 + 1.07i)T^{2} \) |
| 29 | \( 1 + (4.24 + 7.48i)T + (-14.8 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-3.16 - 0.372i)T + (30.1 + 7.20i)T^{2} \) |
| 37 | \( 1 + (-3.72 + 6.95i)T + (-20.4 - 30.8i)T^{2} \) |
| 41 | \( 1 + (1.94 + 0.368i)T + (38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-2.13 + 1.21i)T + (22.0 - 36.8i)T^{2} \) |
| 47 | \( 1 + (-9.96 - 2.87i)T + (39.7 + 25.0i)T^{2} \) |
| 53 | \( 1 + (5.13 + 7.00i)T + (-15.9 + 50.5i)T^{2} \) |
| 59 | \( 1 + (-1.32 - 3.36i)T + (-43.1 + 40.2i)T^{2} \) |
| 61 | \( 1 + (-5.85 - 7.23i)T + (-12.7 + 59.6i)T^{2} \) |
| 67 | \( 1 + (2.45 + 0.966i)T + (49.0 + 45.6i)T^{2} \) |
| 71 | \( 1 + (-3.25 + 4.66i)T + (-24.4 - 66.6i)T^{2} \) |
| 73 | \( 1 + (-3.48 + 9.49i)T + (-55.6 - 47.2i)T^{2} \) |
| 79 | \( 1 + (2.89 + 2.22i)T + (20.1 + 76.3i)T^{2} \) |
| 83 | \( 1 + (1.59 + 16.9i)T + (-81.5 + 15.4i)T^{2} \) |
| 89 | \( 1 + (-1.09 - 6.60i)T + (-84.2 + 28.6i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 1.07i)T + (-20.3 - 94.8i)T^{2} \) |
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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
−11.20115283761584145284936982606, −9.513581416904266318058227832198, −9.116663730968845359093888466719, −8.702844788240429849466397082896, −7.49130305566079557105451755453, −6.15547444704976669579265884475, −4.98951943434619601423406850163, −4.15742498510686828762704133521, −2.45434203971192553439981055752, −1.91111587342730472084300820732,
0.969064439609761032719147990020, 3.09438439911888905429143912854, 3.99842176828936225943741763163, 5.30207440443958534585414153450, 6.36668631939220063324033012079, 7.27529428921014756817024427639, 8.154696757542749368890199779753, 8.662602886100140163300788377982, 9.896014843895020532362659384752, 10.84885147835766412937316808385
