| L(s) = 1 | + (−22.5 + 39.0i)3-s + (184. + 318. i)5-s + (2.04e3 + 6.01e3i)7-s + (8.82e3 + 1.52e4i)9-s + (−3.23e4 + 5.59e4i)11-s − 1.77e5·13-s − 1.65e4·15-s + (2.51e5 − 4.34e5i)17-s + (2.26e5 + 3.92e5i)19-s + (−2.80e5 − 5.57e4i)21-s + (1.10e6 + 1.91e6i)23-s + (9.08e5 − 1.57e6i)25-s − 1.68e6·27-s − 3.33e6·29-s + (5.80e5 − 1.00e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.160 + 0.278i)3-s + (0.131 + 0.228i)5-s + (0.321 + 0.946i)7-s + (0.448 + 0.776i)9-s + (−0.665 + 1.15i)11-s − 1.72·13-s − 0.0845·15-s + (0.729 − 1.26i)17-s + (0.398 + 0.690i)19-s + (−0.315 − 0.0625i)21-s + (0.822 + 1.42i)23-s + (0.465 − 0.805i)25-s − 0.609·27-s − 0.876·29-s + (0.112 − 0.195i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9723528629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9723528629\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.04e3 - 6.01e3i)T \) |
| good | 3 | \( 1 + (22.5 - 39.0i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-184. - 318. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.23e4 - 5.59e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.77e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-2.51e5 + 4.34e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.26e5 - 3.92e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.10e6 - 1.91e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.33e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-5.80e5 + 1.00e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-9.30e6 - 1.61e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.36e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.49e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.86e7 + 4.96e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-1.90e7 + 3.29e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.38e7 + 4.12e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.51e7 + 1.30e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (6.62e7 - 1.14e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 5.78e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-4.32e7 + 7.49e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.25e8 + 2.17e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 4.40e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-9.42e6 - 1.63e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 7.47e8T + 7.60e17T^{2} \) |
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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
−12.26653114934003740709054879053, −11.50075235446548469714400402749, −9.976875897529578536545297331025, −9.663477802801601580962778494239, −7.87460641156421453144351741052, −7.16897289219257714284821550947, −5.26203323310356380364666823008, −4.88669886975357918331191907870, −2.83269474340738242269265836625, −1.84408105982263562290565294835,
0.25862016996642981486764035382, 1.22629373250227278372016389326, 2.95381694362796981856230714054, 4.37110112802178247175593276230, 5.60360011451069894048937378929, 6.96735424622848412322084979713, 7.82456070592640766881230163186, 9.174614827880105616316965992985, 10.30976159011092903701556507897, 11.19531927691419092825111768406
