L(s) = 1 | + (−2 − 3.46i)2-s + (−3.99 + 6.92i)4-s + (−2.5 + 4.33i)5-s + (−3 − 5.19i)7-s + 20·10-s + (16 + 27.7i)11-s + (19 − 32.9i)13-s + (−12 + 20.7i)14-s + (31.9 + 55.4i)16-s − 26·17-s + 100·19-s + (−20.0 − 34.6i)20-s + (63.9 − 110. i)22-s + (−39 + 67.5i)23-s + (−12.5 − 21.6i)25-s − 152·26-s + ⋯ |
L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.223 + 0.387i)5-s + (−0.161 − 0.280i)7-s + 0.632·10-s + (0.438 + 0.759i)11-s + (0.405 − 0.702i)13-s + (−0.229 + 0.396i)14-s + (0.499 + 0.866i)16-s − 0.370·17-s + 1.20·19-s + (−0.223 − 0.387i)20-s + (0.620 − 1.07i)22-s + (−0.353 + 0.612i)23-s + (−0.100 − 0.173i)25-s − 1.14·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9763671364\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9763671364\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (2 + 3.46i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (3 + 5.19i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-16 - 27.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-19 + 32.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 26T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + (39 - 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (25 + 43.3i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-54 + 93.5i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 266T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-11 + 19.0i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (221 + 382. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (257 + 445. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 2T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-250 + 433. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-259 - 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (63 - 109. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 412T + 3.57e5T^{2} \) |
| 73 | \( 1 + 878T + 3.89e5T^{2} \) |
| 79 | \( 1 + (300 + 519. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-141 - 244. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 150T + 7.04e5T^{2} \) |
| 97 | \( 1 + (193 + 334. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.28058560297404711309553622042, −9.926298345251979881918537188879, −8.936969366337035781244683183477, −7.906300266196755379791121207380, −6.90738431130613466228408742885, −5.63933536785303956169027539964, −4.03094424564287645589891679270, −3.11680274847789275431131654342, −1.85873732817549221747354452230, −0.51865190468414100090190551256,
1.04106971645300358767896784375, 3.11120197277054295144879417692, 4.58768982641429185714102578544, 5.85201848454111657624298825277, 6.50280654379135982273855093085, 7.55394054550925843426381606829, 8.425052020578346645661421239904, 9.077107714747055055248797142505, 9.811390665638521358375948076495, 11.24217720848927060810332846948