| L(s) = 1 | − 5.61i·2-s − 23.5·4-s + 11.1i·5-s + 87.4i·8-s + 62.8·10-s + 302.·16-s + 51.3i·17-s − 61.8·19-s − 263. i·20-s + 220. i·23-s − 125.·25-s − 105.·31-s − 1.00e3i·32-s + 288.·34-s + 347. i·38-s + ⋯ |
| L(s) = 1 | − 1.98i·2-s − 2.94·4-s + 0.999i·5-s + 3.86i·8-s + 1.98·10-s + 4.72·16-s + 0.732i·17-s − 0.747·19-s − 2.94i·20-s + 1.99i·23-s − 1.00·25-s − 0.610·31-s − 5.53i·32-s + 1.45·34-s + 1.48i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.763219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.763219\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 11.1iT \) |
| good | 2 | \( 1 + 5.61iT - 8T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 51.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 61.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 220. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 - 545. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 85.1iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 585.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 75.9iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−12.56118495514503708239769932575, −11.50730580515094789302810686176, −10.85615818519175218508840484448, −10.02240903997172058193690178160, −9.065283399789086825112730065768, −7.75101257544140288060404244865, −5.73598349090904568142740630507, −4.12094997205469458270856449969, −3.08947741491031091712916918956, −1.74318480958483440792629824469,
0.40109473387365059637802486695, 4.19952075609584456844054806808, 5.09435691653473785100693386155, 6.20678664300172587326299453825, 7.31586172370153884353975113018, 8.463468560511233810606929903383, 8.999867064618836355856340113530, 10.19489912526372443643586280219, 12.30777100859381738673157832609, 13.08842102181509377643607040434
