L(s) = 1 | + (2 + 2i)2-s − 3.31i·3-s + 8i·4-s + (−1.68 + 1.68i)5-s + (6.62 − 6.62i)6-s + 49.9·7-s + (−16 + 16i)8-s + 70.0·9-s − 6.75·10-s + 233. i·11-s + 26.4·12-s + (101. − 101. i)13-s + (99.8 + 99.8i)14-s + (5.58 + 5.58i)15-s − 64·16-s + (−61.4 + 61.4i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s − 0.367i·3-s + 0.5i·4-s + (−0.0675 + 0.0675i)5-s + (0.183 − 0.183i)6-s + 1.01·7-s + (−0.250 + 0.250i)8-s + 0.864·9-s − 0.0675·10-s + 1.93i·11-s + 0.183·12-s + (0.601 − 0.601i)13-s + (0.509 + 0.509i)14-s + (0.0248 + 0.0248i)15-s − 0.250·16-s + (−0.212 + 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.17395 + 0.975410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17395 + 0.975410i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 2i)T \) |
| 37 | \( 1 + (991. - 944. i)T \) |
good | 3 | \( 1 + 3.31iT - 81T^{2} \) |
| 5 | \( 1 + (1.68 - 1.68i)T - 625iT^{2} \) |
| 7 | \( 1 - 49.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 233. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-101. + 101. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (61.4 - 61.4i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + (-284. + 284. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + (145. - 145. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (26.1 + 26.1i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (1.13e3 + 1.13e3i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + 3.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.14e3 + 2.14e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 1.82e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.21e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (3.79e3 - 3.79e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (2.05e3 + 2.05e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 - 4.31e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.16e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 1.25e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-614. + 614. i)T - 3.89e7iT^{2} \) |
| 83 | \( 1 - 1.18e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-807. - 807. i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (2.34e3 - 2.34e3i)T - 8.85e7iT^{2} \) |
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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
−13.96515124735604308860243626484, −12.93758122230998844764809964295, −12.09409867662684639557806240657, −10.78283519199622901059460638167, −9.320900239742646771696064889265, −7.67186428944501899671111243759, −7.12775120343918874913039283579, −5.33433124805121975581850238194, −4.14410873504473352233437347480, −1.82359154153499580321544085081,
1.34373344324753864800366262603, 3.49851286255482081567946123878, 4.77775391229015804960186323102, 6.16059597918657948394866135839, 8.000360085573070538493377204414, 9.227143861488013337333246819867, 10.72041250195949801611963198000, 11.30612000514164373590548755679, 12.53958483334001731551326765239, 13.89460892279764410051581667094