| L(s) = 1 | + (−1.72 − 6.43i)2-s + (53.7 − 93.1i)3-s + (183. − 105. i)4-s + (−590. + 590. i)5-s + (−691. − 185. i)6-s + (973. − 3.63e3i)7-s + (−2.20e3 − 2.20e3i)8-s + (−2.50e3 − 4.33e3i)9-s + (4.81e3 + 2.78e3i)10-s + (−1.36e4 + 3.64e3i)11-s − 2.27e4i·12-s + (2.64e4 + 1.07e4i)13-s − 2.50e4·14-s + (2.32e4 + 8.67e4i)15-s + (1.67e4 − 2.89e4i)16-s + (1.77e4 − 1.02e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.107 − 0.402i)2-s + (0.663 − 1.14i)3-s + (0.715 − 0.413i)4-s + (−0.945 + 0.945i)5-s + (−0.533 − 0.143i)6-s + (0.405 − 1.51i)7-s + (−0.537 − 0.537i)8-s + (−0.381 − 0.660i)9-s + (0.481 + 0.278i)10-s + (−0.930 + 0.249i)11-s − 1.09i·12-s + (0.926 + 0.375i)13-s − 0.651·14-s + (0.459 + 1.71i)15-s + (0.255 − 0.441i)16-s + (0.213 − 0.122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.977539 - 1.51328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.977539 - 1.51328i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.64e4 - 1.07e4i)T \) |
| good | 2 | \( 1 + (1.72 + 6.43i)T + (-221. + 128i)T^{2} \) |
| 3 | \( 1 + (-53.7 + 93.1i)T + (-3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (590. - 590. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + (-973. + 3.63e3i)T + (-4.99e6 - 2.88e6i)T^{2} \) |
| 11 | \( 1 + (1.36e4 - 3.64e3i)T + (1.85e8 - 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-1.77e4 + 1.02e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.41e5 - 3.79e4i)T + (1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-3.11e5 - 1.79e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.68e5 + 6.37e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.58e5 - 8.58e5i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (-5.60e5 + 1.50e5i)T + (3.04e12 - 1.75e12i)T^{2} \) |
| 41 | \( 1 + (2.83e4 + 1.05e5i)T + (-6.91e12 + 3.99e12i)T^{2} \) |
| 43 | \( 1 + (2.91e6 - 1.68e6i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.89e5 - 1.89e5i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + 3.16e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (2.18e6 - 8.15e6i)T + (-1.27e14 - 7.34e13i)T^{2} \) |
| 61 | \( 1 + (-2.30e6 - 3.99e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.83e6 - 2.92e7i)T + (-3.51e14 + 2.03e14i)T^{2} \) |
| 71 | \( 1 + (1.80e7 + 4.84e6i)T + (5.59e14 + 3.22e14i)T^{2} \) |
| 73 | \( 1 + (1.44e7 + 1.44e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 2.50e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-4.69e7 + 4.69e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + (4.43e7 - 1.18e7i)T + (3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (-6.22e7 - 1.66e7i)T + (6.78e15 + 3.91e15i)T^{2} \) |
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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
−18.16480470327196697474431219283, −15.98803007824779702351950781780, −14.55142473040020172781734245821, −13.36456256414402930322358130778, −11.53922641182021444385988999043, −10.50672650240814818749723769139, −7.66752787385439629348005437517, −7.02996643143243647375634741719, −3.20072590956613686010989019659, −1.21341780776444145119685972294,
3.11712228003096357033467518142, 5.22914444053861086830247084591, 8.143027519291241784023155661947, 8.893892253581062805679419435463, 11.19193640427278475645440823521, 12.53687340724505071775901578070, 15.03136845526629870761439837634, 15.69487097161368559753419870661, 16.34009328074412488262505073817, 18.39621099106414703059761952601
