| L(s) = 1 | + (8 + 13.8i)2-s + (−116. + 201. i)3-s + (−127. + 221. i)4-s + (178. + 308. i)5-s − 3.72e3·6-s − 4.09e3·8-s + (−1.73e4 − 2.99e4i)9-s + (−2.85e3 + 4.94e3i)10-s + (−3.64e4 + 6.31e4i)11-s + (−2.98e4 − 5.16e4i)12-s + 3.93e4·13-s − 8.31e4·15-s + (−3.27e4 − 5.67e4i)16-s + (2.55e5 − 4.43e5i)17-s + (2.77e5 − 4.79e5i)18-s + (−4.50e5 − 7.79e5i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.830 + 1.43i)3-s + (−0.249 + 0.433i)4-s + (0.127 + 0.221i)5-s − 1.17·6-s − 0.353·8-s + (−0.879 − 1.52i)9-s + (−0.0902 + 0.156i)10-s + (−0.750 + 1.29i)11-s + (−0.415 − 0.719i)12-s + 0.382·13-s − 0.424·15-s + (−0.125 − 0.216i)16-s + (0.742 − 1.28i)17-s + (0.622 − 1.07i)18-s + (−0.792 − 1.37i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.572128 + 0.0363073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572128 + 0.0363073i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (116. - 201. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-178. - 308. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.64e4 - 6.31e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 3.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-2.55e5 + 4.43e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.50e5 + 7.79e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.91e5 + 3.31e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 4.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.96e5 + 6.87e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.61e6 - 1.49e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.34e7 - 4.06e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (1.03e7 - 1.79e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.53e6 + 1.13e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.77e7 + 1.34e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.22e8 - 2.12e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 3.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.56e8 + 2.70e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.20e8 + 2.08e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.50e7 + 6.06e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 6.69e8T + 7.60e17T^{2} \) |
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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
−12.02352951620154752555636847659, −10.94871554677650705730265551114, −10.00516081584525787852792967021, −9.112658773930569100324385878522, −7.45143070914079972536975733702, −6.19138791357380286484549295909, −4.97953088996962982127101229139, −4.42609127690259961404881499688, −2.83249417001157879721520183527, −0.17409113828656555799705887972,
1.02460189842318983258086508974, 1.95536200967152296922828252949, 3.61652503339674133477473093915, 5.67050177384797059621280639570, 5.94451188054806254530605775549, 7.59185549490396757351729469747, 8.604836535013332995193701691689, 10.47054620909378264431229698166, 11.17976754495795281357050425249, 12.28195405852878955416877162947
