| L(s) = 1 | + (0.385 + 0.372i)2-s + (−0.0597 − 1.71i)4-s + (1.95 − 0.487i)5-s + (2.18 + 1.70i)7-s + (1.33 − 1.47i)8-s + (0.935 + 0.539i)10-s + (−0.121 + 3.31i)11-s + (6.36 − 3.10i)13-s + (0.206 + 1.47i)14-s + (−2.35 + 0.164i)16-s + (−3.40 + 1.51i)17-s + (−1.25 − 1.12i)19-s + (−0.951 − 3.31i)20-s + (−1.28 + 1.23i)22-s + (1.79 + 4.94i)23-s + ⋯ |
| L(s) = 1 | + (0.272 + 0.263i)2-s + (−0.0298 − 0.855i)4-s + (0.874 − 0.218i)5-s + (0.826 + 0.645i)7-s + (0.470 − 0.522i)8-s + (0.295 + 0.170i)10-s + (−0.0365 + 0.999i)11-s + (1.76 − 0.860i)13-s + (0.0552 + 0.393i)14-s + (−0.588 + 0.0411i)16-s + (−0.824 + 0.367i)17-s + (−0.287 − 0.258i)19-s + (−0.212 − 0.742i)20-s + (−0.272 + 0.262i)22-s + (0.375 + 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.38719 - 0.227237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38719 - 0.227237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.121 - 3.31i)T \) |
| good | 2 | \( 1 + (-0.385 - 0.372i)T + (0.0697 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.95 + 0.487i)T + (4.41 - 2.34i)T^{2} \) |
| 7 | \( 1 + (-2.18 - 1.70i)T + (1.69 + 6.79i)T^{2} \) |
| 13 | \( 1 + (-6.36 + 3.10i)T + (8.00 - 10.2i)T^{2} \) |
| 17 | \( 1 + (3.40 - 1.51i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.25 + 1.12i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 4.94i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (5.76 + 0.809i)T + (27.8 + 7.99i)T^{2} \) |
| 31 | \( 1 + (-4.20 + 6.23i)T + (-11.6 - 28.7i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 3.04i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-2.44 + 0.343i)T + (39.4 - 11.3i)T^{2} \) |
| 43 | \( 1 + (0.620 + 0.739i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.37 - 0.327i)T + (46.8 + 3.27i)T^{2} \) |
| 53 | \( 1 + (2.90 + 4.00i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.57 + 7.31i)T + (-25.8 + 53.0i)T^{2} \) |
| 61 | \( 1 + (1.51 - 1.02i)T + (22.8 - 56.5i)T^{2} \) |
| 67 | \( 1 + (0.347 - 1.97i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.24 - 7.29i)T + (-47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (2.83 + 13.3i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (11.0 - 11.4i)T + (-2.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.67 + 5.48i)T + (-51.0 - 65.4i)T^{2} \) |
| 89 | \( 1 + (9.25 - 5.34i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.613 - 2.45i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.01156516823797813123246266645, −9.322871976887879439004111441190, −8.536222115830130452107330038326, −7.50695232786528526396396607652, −6.27450073545466652924681350624, −5.77410716288693469512482525383, −5.02529182750941195724080567802, −4.01356078490652210838587904713, −2.19835390499401292273369059887, −1.37661165630494538786593062342,
1.46738443383074519366010064929, 2.66700535955149603574363319269, 3.86054092917637281037047405765, 4.54946043398526313425409878581, 5.87307632150256583072511900343, 6.63780613293920722255054376739, 7.66855612932194024804812902671, 8.666979797634091036468246387613, 8.977424127486035740543945029809, 10.51125952121739492007242014791
