| L(s) = 1 | + (−8 + 13.8i)2-s + (−135. − 234. i)3-s + (−127. − 221. i)4-s + (−684. + 1.18e3i)5-s + 4.33e3·6-s + (6.32e3 + 558. i)7-s + 4.09e3·8-s + (−2.68e4 + 4.64e4i)9-s + (−1.09e4 − 1.89e4i)10-s + (5.46e3 + 9.46e3i)11-s + (−3.46e4 + 6.00e4i)12-s − 7.96e3·13-s + (−5.83e4 + 8.32e4i)14-s + 3.70e5·15-s + (−3.27e4 + 5.67e4i)16-s + (1.62e5 + 2.80e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.965 − 1.67i)3-s + (−0.249 − 0.433i)4-s + (−0.489 + 0.848i)5-s + 1.36·6-s + (0.996 + 0.0879i)7-s + 0.353·8-s + (−1.36 + 2.36i)9-s + (−0.346 − 0.600i)10-s + (0.112 + 0.194i)11-s + (−0.482 + 0.835i)12-s − 0.0773·13-s + (−0.406 + 0.578i)14-s + 1.89·15-s + (−0.125 + 0.216i)16-s + (0.470 + 0.814i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.425581 + 0.415184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.425581 + 0.415184i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-6.32e3 - 558. i)T \) |
| good | 3 | \( 1 + (135. + 234. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (684. - 1.18e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-5.46e3 - 9.46e3i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 7.96e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.62e5 - 2.80e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.15e5 + 2.00e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.05e6 - 1.82e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.46e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-9.89e5 - 1.71e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (2.38e6 - 4.12e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 5.57e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.15e6 - 3.73e6i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.51e7 - 7.81e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.64e7 - 6.31e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-3.71e6 + 6.43e6i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (5.28e7 + 9.16e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.02e8 + 1.77e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (4.43e7 - 7.68e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.74e8 - 4.75e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.16e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.85824879040509366928867904612, −16.91250388407248554845011034408, −14.97268849473940675044974888481, −13.62331568162342523353038628067, −11.94233830696205001687561364272, −10.92640872431146613271049733457, −7.962343782178611885070074950542, −7.11057869757295200526171999191, −5.61581954701456905887601467556, −1.55965868580082390688896143531,
0.42995120715677560023629408979, 4.04109437924163289481913412426, 5.18477611890326861113988636492, 8.538156081920978798464806868610, 9.939952173998604861657109366882, 11.24575387353640615698015703280, 12.09407278714816045956289049875, 14.64468011852755059080377934666, 16.19017703883084557510471227059, 16.84777810034635349619724540376
