| L(s) = 1 | + (3.23 + 2.35i)2-s + (5.47 + 16.8i)3-s + (4.94 + 15.2i)4-s + (53.8 + 15.0i)5-s + (−21.8 + 67.3i)6-s − 2.11·7-s + (−19.7 + 60.8i)8-s + (−57.1 + 41.5i)9-s + (138. + 175. i)10-s + (−109. − 79.3i)11-s + (−229. + 166. i)12-s + (−488. + 355. i)13-s + (−6.83 − 4.96i)14-s + (41.8 + 989. i)15-s + (−207. + 150. i)16-s + (160. − 493. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.351 + 1.08i)3-s + (0.154 + 0.475i)4-s + (0.963 + 0.268i)5-s + (−0.248 + 0.764i)6-s − 0.0162·7-s + (−0.109 + 0.336i)8-s + (−0.235 + 0.170i)9-s + (0.439 + 0.553i)10-s + (−0.272 − 0.197i)11-s + (−0.459 + 0.333i)12-s + (−0.802 + 0.582i)13-s + (−0.00931 − 0.00676i)14-s + (0.0479 + 1.13i)15-s + (−0.202 + 0.146i)16-s + (0.134 − 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.68020 + 2.21525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68020 + 2.21525i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 5 | \( 1 + (-53.8 - 15.0i)T \) |
| good | 3 | \( 1 + (-5.47 - 16.8i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 2.11T + 1.68e4T^{2} \) |
| 11 | \( 1 + (109. + 79.3i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (488. - 355. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-160. + 493. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-232. + 715. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.41e3 - 1.02e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.36e3 + 7.29e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-5.00 + 15.3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.23e3 + 4.52e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.39e4 + 1.01e4i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 6.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (418. + 1.28e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.93e3 + 5.94e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.05e4 - 2.21e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.35e4 - 1.71e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.03e4 - 3.18e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.30e4 - 7.09e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.67e4 + 1.94e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (9.01e3 + 2.77e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.62e4 + 4.99e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (9.25e4 + 6.72e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.95e4 + 1.52e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.82045752368304894567150279554, −14.02891672045019754268192628208, −12.91737717897401919882914878304, −11.29161572726656494808336419719, −9.927766566552924449337231698123, −9.119210474516214479207147247612, −7.23701585372992079244470173272, −5.67264733735655395067403109613, −4.37436623269746871668752094851, −2.69893906330733390334376027273,
1.36046404444931278999813226155, 2.68549938866196464546110833198, 5.05364484995672793422520325362, 6.47056283538929496220296989780, 7.85986689240464063375561744805, 9.508686458904838185548421365396, 10.72931093661218310407412486872, 12.58231836529103343581752229195, 12.80623338185086789525685126463, 13.98320062612926331455797827651
