| L(s) = 1 | + (−4.87 + 1.80i)3-s − 8.95i·5-s + (20.4 − 17.5i)9-s + 36.4i·11-s + (16.1 + 43.6i)15-s − 2.65i·23-s + 44.8·25-s + (−68 + 122. i)27-s − 118.·31-s + (−65.8 − 177. i)33-s − 113.·37-s + (−157. − 183. i)45-s − 643. i·47-s + 343·49-s + 225. i·53-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.347i)3-s − 0.800i·5-s + (0.758 − 0.651i)9-s + 1.00i·11-s + (0.278 + 0.751i)15-s − 0.0240i·23-s + 0.358·25-s + (−0.484 + 0.874i)27-s − 0.685·31-s + (−0.347 − 0.937i)33-s − 0.504·37-s + (−0.522 − 0.607i)45-s − 1.99i·47-s + 49-s + 0.584i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.059155293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059155293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.87 - 1.80i)T \) |
| 11 | \( 1 - 36.4iT \) |
| good | 5 | \( 1 + 8.95iT - 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 2.65iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 113.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 643. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 225. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 898. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.08e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.67e3T + 9.12e5T^{2} \) |
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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
−10.27023864245671738958302273187, −9.508610456554598278117135769032, −8.659924815640582356603519751145, −7.40857997492150532925498939323, −6.55628365082069979200652743241, −5.36937206770353546404721977701, −4.77317315973740750776589319256, −3.73304740175502106405738886968, −1.82493939414226015478476299289, −0.45309091328115380591277767427,
1.01564968400127826543567921057, 2.58524513078746341335265009706, 3.87481588122983611703504307194, 5.21743505059427081095138883941, 6.06930515550060915144682174023, 6.86194953369161030691005768824, 7.69634286616344508527303091984, 8.808376165511382234226047582128, 10.00910707106668938243034446384, 10.85380147198457814534163503123
