L(s) = 1 | + (2.36 + 1.54i)2-s + (3.20 + 7.32i)4-s − 1.43i·5-s − 27.5i·7-s + (−3.76 + 22.3i)8-s + (2.21 − 3.38i)10-s + 22.2·11-s + 69.1·13-s + (42.6 − 65.2i)14-s + (−43.4 + 46.9i)16-s − 31.4i·17-s + 11.4i·19-s + (10.4 − 4.58i)20-s + (52.5 + 34.3i)22-s + 145.·23-s + ⋯ |
L(s) = 1 | + (0.836 + 0.547i)2-s + (0.400 + 0.916i)4-s − 0.127i·5-s − 1.48i·7-s + (−0.166 + 0.986i)8-s + (0.0700 − 0.107i)10-s + 0.608·11-s + 1.47·13-s + (0.815 − 1.24i)14-s + (−0.678 + 0.734i)16-s − 0.448i·17-s + 0.138i·19-s + (0.117 − 0.0512i)20-s + (0.509 + 0.333i)22-s + 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.344706134\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.344706134\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.36 - 1.54i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.43iT - 125T^{2} \) |
| 7 | \( 1 + 27.5iT - 343T^{2} \) |
| 11 | \( 1 - 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 11.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 118. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 398. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 200. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 83.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 398.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 355. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 64.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 409. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−11.18341660687113144489985548044, −10.70052327002637214472328899383, −9.168549064712854763800426959829, −8.209269705526057103927792256250, −7.08472102228889632331975603295, −6.53470816386785019757896268256, −5.17247588772453002493650037884, −4.10173699108962996084667399746, −3.28245726980162742154119198924, −1.16367271415762390994403779742,
1.35344158497461447223642431707, 2.70468421742681687866187353449, 3.77612312438022450873627910616, 5.15272273528354825397912237935, 6.00369684355986345857195235704, 6.85507780258756236082490753130, 8.665572301666962316129763069028, 9.151729061017908126442050221009, 10.55837296053137042962353198696, 11.23084188361154283426360823426