| L(s) = 1 | + (−123. + 213. i)3-s + (1.00e3 + 1.74e3i)5-s + (−2.06e4 − 3.58e4i)9-s + (2.39e4 − 4.15e4i)11-s + 1.71e5·13-s − 4.97e5·15-s + (1.67e5 − 2.89e5i)17-s + (−9.63e3 − 1.66e4i)19-s + (6.62e5 + 1.14e6i)23-s + (−1.05e6 + 1.82e6i)25-s + 5.35e6·27-s − 1.11e6·29-s + (−4.65e6 + 8.05e6i)31-s + (5.92e6 + 1.02e7i)33-s + (8.46e6 + 1.46e7i)37-s + ⋯ |
| L(s) = 1 | + (−0.880 + 1.52i)3-s + (0.720 + 1.24i)5-s + (−1.05 − 1.81i)9-s + (0.494 − 0.855i)11-s + 1.66·13-s − 2.53·15-s + (0.486 − 0.842i)17-s + (−0.0169 − 0.0293i)19-s + (0.493 + 0.855i)23-s + (−0.538 + 0.931i)25-s + 1.93·27-s − 0.294·29-s + (−0.904 + 1.56i)31-s + (0.870 + 1.50i)33-s + (0.742 + 1.28i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.902871194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902871194\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (123. - 213. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.00e3 - 1.74e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-2.39e4 + 4.15e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.71e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.67e5 + 2.89e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (9.63e3 + 1.66e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-6.62e5 - 1.14e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.11e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (4.65e6 - 8.05e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.46e6 - 1.46e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 9.85e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.62e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.34e7 - 2.33e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.12e7 + 7.13e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.53e7 + 1.13e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.79e7 - 8.30e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.51e6 + 9.54e6i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.60e8 - 2.77e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.94e7 + 3.36e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.66e8 - 4.60e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.18e9T + 7.60e17T^{2} \) |
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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
−11.16540541523993420516900167795, −10.42268292227867695874492616106, −9.618341542673337953664372357069, −8.689210821596371794163198289765, −6.80301237817410429884032263992, −5.99365212897210611191562919946, −5.23137299349518649807631515851, −3.69779266353331047777595719221, −3.13031836658044444730239564295, −1.10580402805226663280400554854,
0.54879930211477843169362863672, 1.34677460558024951426108735865, 2.01240158240022863122412819789, 4.20856829130029921321797049101, 5.63179018633053186647037822770, 6.05348119845853497623302676328, 7.21288601455843923383870809746, 8.314947942388317503354100356029, 9.185898294922602303538481077181, 10.61324191975706147373989136345
