L(s) = 1 | + (1.52 + 0.814i)3-s + (2.08 − 2.08i)5-s − 1.14·7-s + (1.67 + 2.48i)9-s + (1.67 + 1.67i)11-s + (−0.146 + 0.146i)13-s + (4.88 − 1.48i)15-s − 5.59i·17-s + (−1.48 − 1.48i)19-s + (−1.75 − 0.933i)21-s + 3.34i·23-s − 3.68i·25-s + (0.533 + 5.16i)27-s + (−3.51 − 3.51i)29-s + 5.83i·31-s + ⋯ |
L(s) = 1 | + (0.882 + 0.470i)3-s + (0.931 − 0.931i)5-s − 0.433·7-s + (0.558 + 0.829i)9-s + (0.504 + 0.504i)11-s + (−0.0405 + 0.0405i)13-s + (1.26 − 0.384i)15-s − 1.35i·17-s + (−0.341 − 0.341i)19-s + (−0.382 − 0.203i)21-s + 0.698i·23-s − 0.737i·25-s + (0.102 + 0.994i)27-s + (−0.652 − 0.652i)29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00865 + 0.0489117i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00865 + 0.0489117i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.52 - 0.814i)T \) |
good | 5 | \( 1 + (-2.08 + 2.08i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + (-1.67 - 1.67i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.146 - 0.146i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.59iT - 17T^{2} \) |
| 19 | \( 1 + (1.48 + 1.48i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.34iT - 23T^{2} \) |
| 29 | \( 1 + (3.51 + 3.51i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.83iT - 31T^{2} \) |
| 37 | \( 1 + (-4.83 - 4.83i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.610T + 41T^{2} \) |
| 43 | \( 1 + (-1.48 + 1.48i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 + (-0.164 + 0.164i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.05 + 9.05i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.53 - 4.53i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.635 - 0.635i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.90iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 9.83iT - 79T^{2} \) |
| 83 | \( 1 + (8.09 - 8.09i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.490T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−11.29766654609064562349428281583, −9.966099965868296919249950717585, −9.473901592580769185314901363141, −8.926620466076312570824926947220, −7.76350859467104471307339413909, −6.63719472092198756302100916060, −5.27808263481567227294316897803, −4.46689204175197002524389802077, −3.04602305193807420428062307323, −1.69902914385269716382626458823,
1.80122298426281180647347900878, 2.93724514502245192657080780505, 3.99006081777247294808988599521, 6.04982151947964235625684984650, 6.42883560351531077229758033043, 7.58516964201251407175194706823, 8.611449287230137118370234007666, 9.505263149989931049146510960775, 10.28108618760217909533252741465, 11.17688657475679132793741291059