L(s) = 1 | + (1.88 − 0.391i)2-s + (−0.730 + 0.682i)3-s + (2.48 − 1.07i)4-s + (−1.11 + 1.57i)6-s + (2.68 − 1.89i)8-s + (0.0682 − 0.997i)9-s + (−1.07 + 2.48i)12-s + (2.47 − 2.65i)16-s + (−1.46 − 0.519i)17-s + (−0.262 − 1.90i)18-s + (1.76 + 0.494i)19-s + (0.398 + 0.0827i)23-s + (−0.669 + 3.22i)24-s + (0.631 + 0.775i)27-s + (−0.457 + 0.489i)31-s + (1.92 − 3.15i)32-s + ⋯ |
L(s) = 1 | + (1.88 − 0.391i)2-s + (−0.730 + 0.682i)3-s + (2.48 − 1.07i)4-s + (−1.11 + 1.57i)6-s + (2.68 − 1.89i)8-s + (0.0682 − 0.997i)9-s + (−1.07 + 2.48i)12-s + (2.47 − 2.65i)16-s + (−1.46 − 0.519i)17-s + (−0.262 − 1.90i)18-s + (1.76 + 0.494i)19-s + (0.398 + 0.0827i)23-s + (−0.669 + 3.22i)24-s + (0.631 + 0.775i)27-s + (−0.457 + 0.489i)31-s + (1.92 − 3.15i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.339421124\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.339421124\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.730 - 0.682i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.887 + 0.460i)T \) |
good | 2 | \( 1 + (-1.88 + 0.391i)T + (0.917 - 0.398i)T^{2} \) |
| 7 | \( 1 + (0.203 + 0.979i)T^{2} \) |
| 11 | \( 1 + (0.775 - 0.631i)T^{2} \) |
| 13 | \( 1 + (0.962 + 0.269i)T^{2} \) |
| 17 | \( 1 + (1.46 + 0.519i)T + (0.775 + 0.631i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 0.494i)T + (0.854 + 0.519i)T^{2} \) |
| 23 | \( 1 + (-0.398 - 0.0827i)T + (0.917 + 0.398i)T^{2} \) |
| 29 | \( 1 + (-0.962 + 0.269i)T^{2} \) |
| 31 | \( 1 + (0.457 - 0.489i)T + (-0.0682 - 0.997i)T^{2} \) |
| 37 | \( 1 + (-0.576 + 0.816i)T^{2} \) |
| 41 | \( 1 + (0.334 + 0.942i)T^{2} \) |
| 43 | \( 1 + (0.682 - 0.730i)T^{2} \) |
| 53 | \( 1 + (-0.111 - 0.0787i)T + (0.334 + 0.942i)T^{2} \) |
| 59 | \( 1 + (-0.682 - 0.730i)T^{2} \) |
| 61 | \( 1 + (0.911 - 1.75i)T + (-0.576 - 0.816i)T^{2} \) |
| 67 | \( 1 + (0.203 - 0.979i)T^{2} \) |
| 71 | \( 1 + (0.917 + 0.398i)T^{2} \) |
| 73 | \( 1 + (-0.990 + 0.136i)T^{2} \) |
| 79 | \( 1 + (-0.572 - 0.347i)T + (0.460 + 0.887i)T^{2} \) |
| 83 | \( 1 + (1.08 - 0.386i)T + (0.775 - 0.631i)T^{2} \) |
| 89 | \( 1 + (-0.854 + 0.519i)T^{2} \) |
| 97 | \( 1 + (-0.0682 + 0.997i)T^{2} \) |
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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
−8.924609868700771895039462693559, −7.47652154895210747505169145380, −6.83319298917301377308212355866, −6.18617273165154566660890361306, −5.29467953405950355202406412703, −5.02143544600188493601697054922, −4.13180556033075321093927664415, −3.46293804511208225499954476570, −2.65239077771426348287083405782, −1.36372209337494026140227324757,
1.59983690087475078512271061558, 2.56147226166816236979546493963, 3.43430882855498225422858555338, 4.54174711432677563889727630952, 4.95846949477740054686363876679, 5.81339280838774454824969510211, 6.33907985813749284822173164594, 7.06663824429226091888645477279, 7.53117745388159816672271059944, 8.370109716393518159774398020511