| L(s) = 1 | + (3.23 − 2.35i)2-s − 2.80i·3-s + (4.94 − 15.2i)4-s + (−7.78 + 23.9i)5-s + (−6.58 − 9.06i)6-s + (−138. + 189. i)7-s + (−19.7 − 60.8i)8-s + 235.·9-s + (31.1 + 95.8i)10-s + (−705. + 229. i)11-s + (−42.6 − 13.8i)12-s + (317. + 437. i)13-s + 939. i·14-s + (67.0 + 21.7i)15-s + (−207. − 150. i)16-s + (1.20e3 − 392. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s − 0.179i·3-s + (0.154 − 0.475i)4-s + (−0.139 + 0.428i)5-s + (−0.0746 − 0.102i)6-s + (−1.06 + 1.46i)7-s + (−0.109 − 0.336i)8-s + 0.967·9-s + (0.0984 + 0.302i)10-s + (−1.75 + 0.571i)11-s + (−0.0854 − 0.0277i)12-s + (0.521 + 0.717i)13-s + 1.28i·14-s + (0.0769 + 0.0250i)15-s + (−0.202 − 0.146i)16-s + (1.01 − 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.13679 + 0.994610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13679 + 0.994610i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 41 | \( 1 + (714. - 1.07e4i)T \) |
| good | 3 | \( 1 + 2.80iT - 243T^{2} \) |
| 5 | \( 1 + (7.78 - 23.9i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (138. - 189. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (705. - 229. i)T + (1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-317. - 437. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.20e3 + 392. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (234. - 322. i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (2.98e3 - 2.16e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.88e3 + 611. i)T + (1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-535. - 1.64e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.24e3 + 3.82e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 43 | \( 1 + (911. - 661. i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-359. - 494. i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.56e4 + 8.34e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.77e4 + 2.74e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.69e4 + 1.23e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-9.45e3 - 3.07e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.56e4 - 8.33e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + 1.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.48e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 5.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.37e4 - 3.27e4i)T + (-1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.69e5 - 5.51e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.21614606314623927753661734776, −12.66243951381028079941367783616, −11.70379928637331859769886664913, −10.26669537283218598032586538061, −9.477994755844923181713218997453, −7.72065492057618777451411373605, −6.35915268877349945168603402024, −5.19039359415823523081251745147, −3.36170301503651861987657770719, −2.10537001262170126958769765543,
0.51045295528027533840680397546, 3.26048698149957952894786558629, 4.42270497528882175361410096600, 5.89027570324208595472425285698, 7.27995418354318673569215681403, 8.195545310156914801220920013325, 10.12874204260796715942408443427, 10.59488172420874834343825057027, 12.61907699966111563685898406971, 13.06882432982823014341939351697
