| L(s) = 1 | + (−1 + 1.73i)3-s + (14 − 12.1i)7-s + (11.5 + 19.9i)9-s + (−13.5 + 23.3i)11-s − 41·13-s + (3 − 5.19i)17-s + (24.5 + 42.4i)19-s + (7 + 36.3i)21-s + (−40.5 − 70.1i)23-s − 100·27-s + 66·29-s + (−94 + 162. i)31-s + (−27 − 46.7i)33-s + (11.5 + 19.9i)37-s + (41 − 71.0i)39-s + ⋯ |
| L(s) = 1 | + (−0.192 + 0.333i)3-s + (0.755 − 0.654i)7-s + (0.425 + 0.737i)9-s + (−0.370 + 0.640i)11-s − 0.874·13-s + (0.0428 − 0.0741i)17-s + (0.295 + 0.512i)19-s + (0.0727 + 0.377i)21-s + (−0.367 − 0.635i)23-s − 0.712·27-s + 0.422·29-s + (−0.544 + 0.943i)31-s + (−0.142 − 0.246i)33-s + (0.0510 + 0.0885i)37-s + (0.168 − 0.291i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.225490756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225490756\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-14 + 12.1i)T \) |
| good | 3 | \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \) |
| 11 | \( 1 + (13.5 - 23.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 41T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-24.5 - 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (40.5 + 70.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 66T + 2.43e4T^{2} \) |
| 31 | \( 1 + (94 - 162. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-11.5 - 19.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 51T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-163.5 - 283. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (100.5 - 174. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-12 + 20.7i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-98 - 169. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (482 - 834. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (434 - 751. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (319 + 552. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 36T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-405 - 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−10.34979874848305794515212514826, −9.786631590211318691661371632496, −8.552792545653409106055251587588, −7.57151783707506605094338842154, −7.16850401058662383497475199172, −5.67499983206363032441658796992, −4.75461670455121772192526299573, −4.19792053258464643542156809269, −2.58245930533965107306292880472, −1.39654611794803233781750157505,
0.34980073621966933635382921841, 1.73119121991548992887474259626, 2.92147241796739193179961975788, 4.24691272149273105435321020939, 5.33768252040448818706634074621, 6.07323850817422622056525166692, 7.22296399279699807822996801948, 7.900114619198913075860363100157, 8.945122767897129582860551465108, 9.633502034282451949391886512691
