| L(s) = 1 | + 1.73i·5-s − 3i·7-s − 5.19·11-s + 13-s + 3.46i·17-s + 6i·19-s + 5.19·23-s + 2.00·25-s + 8.66i·29-s + 3i·31-s + 5.19·35-s − 4·37-s + 5.19i·41-s − 3i·43-s + 5.19·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s − 1.13i·7-s − 1.56·11-s + 0.277·13-s + 0.840i·17-s + 1.37i·19-s + 1.08·23-s + 0.400·25-s + 1.60i·29-s + 0.538i·31-s + 0.878·35-s − 0.657·37-s + 0.811i·41-s − 0.457i·43-s + 0.757·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.168973905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168973905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 8.66iT - 29T^{2} \) |
| 31 | \( 1 - 3iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 15iT - 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−10.28022356976502912246609958547, −9.011104600844225295069404063802, −8.059077091770750891174460444435, −7.41036050568957028164075060786, −6.72017047940997203420020949378, −5.70383470913063902782134990027, −4.76587075501338825507301666772, −3.62078361246027658319344766399, −2.89775493735339319923820077677, −1.39537144777438267960550015576,
0.50023974819583373243254670989, 2.29036642415981658647050751702, 2.98399805629499848359794861927, 4.62604249616783951873412764168, 5.16948438981566945141074046682, 5.89956498832381126244273500963, 7.09016961595107276830548253621, 7.931538569950578043762027606676, 8.825973834413838617411567396696, 9.176585378677702784303660228332
