L(s) = 1 | + (−2.16 + 3.75i)2-s + (−0.193 − 5.19i)3-s + (−5.39 − 9.34i)4-s + (19.9 + 10.5i)6-s + (−6.50 + 11.2i)7-s + 12.1·8-s + (−26.9 + 2.01i)9-s + (17.2 − 29.9i)11-s + (−47.5 + 29.8i)12-s + (3.77 + 6.54i)13-s + (−28.1 − 48.8i)14-s + (16.9 − 29.2i)16-s − 82.0·17-s + (50.8 − 105. i)18-s + 146.·19-s + ⋯ |
L(s) = 1 | + (−0.766 + 1.32i)2-s + (−0.0373 − 0.999i)3-s + (−0.674 − 1.16i)4-s + (1.35 + 0.716i)6-s + (−0.351 + 0.607i)7-s + 0.535·8-s + (−0.997 + 0.0745i)9-s + (0.473 − 0.820i)11-s + (−1.14 + 0.717i)12-s + (0.0806 + 0.139i)13-s + (−0.538 − 0.931i)14-s + (0.264 − 0.457i)16-s − 1.17·17-s + (0.665 − 1.38i)18-s + 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0997 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0997 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.683778 + 0.618680i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.683778 + 0.618680i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.193 + 5.19i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (2.16 - 3.75i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (6.50 - 11.2i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-17.2 + 29.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-3.77 - 6.54i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 82.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-93.9 - 162. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (21.8 - 37.8i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-29.3 - 50.8i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 329.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-86.4 - 149. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-78.3 + 135. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-84.6 + 146. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-297. - 514. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (162. - 281. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-344. - 596. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 515.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-262. + 455. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-179. + 310. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-197. + 342. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.94541021021411003418279694915, −11.21711760284167656782455440438, −9.387776611485774380544283546147, −8.954400518927496232259648470177, −7.86516019618231113038936145265, −7.06117262967820398186230592738, −6.15946976242015277679681677840, −5.40146085439402333411826415386, −3.01881461697562790687190688027, −0.999672267602924492798341047880,
0.65800697878304793198026705792, 2.52790187606086896147840763034, 3.70299164179570601901153681764, 4.74887608915904820629419673524, 6.49083535301710403819738657757, 8.010114456956847109242658410866, 9.369735460590859693545509392454, 9.524111548470367676781497237820, 10.66217933102895790519669181550, 11.19180182548546896843852370232