L(s) = 1 | + (3.64 + 2.10i)2-s + (4.80 − 1.98i)3-s + (4.87 + 8.44i)4-s + (−2.5 + 4.33i)5-s + (21.7 + 2.86i)6-s + (16.7 − 7.79i)7-s + 7.40i·8-s + (19.1 − 19.0i)9-s + (−18.2 + 10.5i)10-s + (−50.2 + 28.9i)11-s + (40.2 + 30.8i)12-s + 47.9i·13-s + (77.7 + 6.94i)14-s + (−3.39 + 25.7i)15-s + (23.4 − 40.5i)16-s + (−11.9 − 20.6i)17-s + ⋯ |
L(s) = 1 | + (1.29 + 0.744i)2-s + (0.924 − 0.382i)3-s + (0.609 + 1.05i)4-s + (−0.223 + 0.387i)5-s + (1.47 + 0.194i)6-s + (0.907 − 0.420i)7-s + 0.327i·8-s + (0.707 − 0.706i)9-s + (−0.577 + 0.333i)10-s + (−1.37 + 0.794i)11-s + (0.967 + 0.742i)12-s + 1.02i·13-s + (1.48 + 0.132i)14-s + (−0.0585 + 0.443i)15-s + (0.366 − 0.634i)16-s + (−0.169 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.59893 + 1.31378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.59893 + 1.31378i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.80 + 1.98i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (-16.7 + 7.79i)T \) |
good | 2 | \( 1 + (-3.64 - 2.10i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (50.2 - 28.9i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 47.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (11.9 + 20.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (116. + 67.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-84.9 - 49.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 57.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (124. - 71.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (0.379 - 0.657i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 518.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-140. + 243. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-168. + 97.2i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-244. - 423. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-159. - 91.8i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-452. - 783. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 524. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (110. - 64.0i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-120. + 208. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 607.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-436. + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.29e3iT - 9.12e5T^{2} \) |
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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
−13.49841015264979853683926714985, −12.94300247984890429462763897943, −11.66824585749359288223850880931, −10.26799346272087109841372211762, −8.639265676573550719753669010333, −7.37604219436964466879818049397, −6.88188493780785316420827004721, −5.01289426607482103350642135518, −4.02168219124569694515586715952, −2.36154920568173110419459021925,
2.14404990955209998774383314686, 3.36207381872411027417141431960, 4.68715483137897148579273590222, 5.57271367420706600678844806487, 8.016740952996146325648530323885, 8.560497314226886125477460005456, 10.47956329310996391028867536569, 11.01573219934608040061733286941, 12.56482454205194468308340850203, 13.04741370291818392031844606814