| L(s) = 1 | + (−8 − 13.8i)2-s + (85 − 147. i)3-s + (−127. + 221. i)4-s + (272 + 471. i)5-s − 2.72e3·6-s + 4.09e3·8-s + (−4.60e3 − 7.98e3i)9-s + (4.35e3 − 7.53e3i)10-s + (−2.44e4 + 4.22e4i)11-s + (2.17e4 + 3.76e4i)12-s + 1.58e4·13-s + 9.24e4·15-s + (−3.27e4 − 5.67e4i)16-s + (−1.07e4 + 1.85e4i)17-s + (−7.37e4 + 1.27e5i)18-s + (−3.58e5 − 6.20e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.605 − 1.04i)3-s + (−0.249 + 0.433i)4-s + (0.194 + 0.337i)5-s − 0.856·6-s + 0.353·8-s + (−0.234 − 0.405i)9-s + (0.137 − 0.238i)10-s + (−0.502 + 0.870i)11-s + (0.302 + 0.524i)12-s + 0.154·13-s + 0.471·15-s + (−0.125 − 0.216i)16-s + (−0.0310 + 0.0538i)17-s + (−0.165 + 0.286i)18-s + (−0.630 − 1.09i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.73604 - 1.32070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73604 - 1.32070i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-85 + 147. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-272 - 471. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.44e4 - 4.22e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.58e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.07e4 - 1.85e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (3.58e5 + 6.20e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.23e6 - 2.13e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 5.55e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.89e6 + 5.02e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.94e6 - 3.37e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 6.36e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.82e7 - 4.89e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.99e7 + 5.18e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-8.28e7 + 1.43e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-2.57e7 - 4.45e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (4.67e7 - 8.10e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 9.56e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.53e8 + 2.65e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.48e8 + 4.29e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 3.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (8.27e7 + 1.43e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 7.58e8T + 7.60e17T^{2} \) |
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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
−11.99002157214376322502750534899, −10.81244273517452824280705794133, −9.724091350715035346418469053200, −8.533481962835343210836189350695, −7.54284452902635867049470989945, −6.62064455681018898000585962434, −4.70331614376073596831723991258, −2.90646386792010424596404935256, −2.09979459345909241255770644995, −0.853950227704393326447897344025,
0.883514697767720105872092082395, 2.86242646544434583849558682444, 4.24235621477520368486497675920, 5.36824884895012053231788120376, 6.70979904824197983516457868873, 8.444940429818283912121152798605, 8.756149618603340414563188519655, 10.10568917230731210793538943346, 10.71481230770850159051924591987, 12.45330353235755547236486059813
