| L(s) = 1 | + (0.675 + 0.219i)2-s + (1.44 − 1.05i)3-s + (−2.82 − 2.05i)4-s + (1.51 + 4.64i)5-s + (1.20 − 0.392i)6-s + (−5.17 + 7.12i)7-s + (−3.13 − 4.30i)8-s + (−1.79 + 5.51i)9-s + 3.47i·10-s + (−2.65 + 10.6i)11-s − 6.25·12-s + (7.38 + 2.40i)13-s + (−5.05 + 3.67i)14-s + (7.07 + 5.13i)15-s + (3.15 + 9.69i)16-s + (7.23 − 2.35i)17-s + ⋯ |
| L(s) = 1 | + (0.337 + 0.109i)2-s + (0.482 − 0.350i)3-s + (−0.706 − 0.513i)4-s + (0.302 + 0.929i)5-s + (0.201 − 0.0654i)6-s + (−0.739 + 1.01i)7-s + (−0.391 − 0.538i)8-s + (−0.199 + 0.613i)9-s + 0.347i·10-s + (−0.240 + 0.970i)11-s − 0.520·12-s + (0.568 + 0.184i)13-s + (−0.361 + 0.262i)14-s + (0.471 + 0.342i)15-s + (0.196 + 0.606i)16-s + (0.425 − 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0488 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0488 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.990982 + 1.04061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990982 + 1.04061i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (2.65 - 10.6i)T \) |
| 23 | \( 1 + 4.79T \) |
| good | 2 | \( 1 + (-0.675 - 0.219i)T + (3.23 + 2.35i)T^{2} \) |
| 3 | \( 1 + (-1.44 + 1.05i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (-1.51 - 4.64i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (5.17 - 7.12i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-7.38 - 2.40i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-7.23 + 2.35i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (2.74 + 3.78i)T + (-111. + 343. i)T^{2} \) |
| 29 | \( 1 + (3.01 - 4.15i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-1.25 + 3.86i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (12.0 + 8.76i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (10.7 + 14.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 10.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (54.0 - 39.2i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (0.202 - 0.623i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (62.6 + 45.5i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-77.4 + 25.1i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 41.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-23.6 - 72.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (3.57 - 4.91i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-88.9 - 28.9i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-125. + 40.8i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 57.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-22.7 + 70.1i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−12.40938490685175529860275938169, −10.99105120383017387188498784223, −10.00101857910464453079054377055, −9.265031321560848723585432851863, −8.222671016214993784066755271383, −6.89472219461797488333695240952, −6.00235559440981905045977742583, −4.94028907211232758773291241294, −3.30226194190774658881672090694, −2.13201398429845879038364548696,
0.64702534168029926299645752070, 3.27732921779551814231131949353, 3.86619173309545822987493891690, 5.16230634223610519060447231930, 6.34894647646015010485980645009, 7.981148689181903170247083593991, 8.726389262451708997130347608023, 9.478093761782610405203780863500, 10.43559662901108973110726146962, 11.80363062811481378076820668349
