| L(s) = 1 | + (0.909 + 15.9i)2-s + (−113. − 37.0i)3-s + (−254. + 29.0i)4-s + (−151. + 606. i)5-s + (487. − 1.85e3i)6-s + 1.29e3i·7-s + (−695. − 4.03e3i)8-s + (6.29e3 + 4.57e3i)9-s + (−9.82e3 − 1.86e3i)10-s + (−3.33e3 − 4.59e3i)11-s + (3.00e4 + 6.10e3i)12-s + (−3.80e4 − 2.76e4i)13-s + (−2.06e4 + 1.17e3i)14-s + (3.96e4 − 6.34e4i)15-s + (6.38e4 − 1.47e4i)16-s + (−3.76e4 − 1.15e5i)17-s + ⋯ |
| L(s) = 1 | + (0.0568 + 0.998i)2-s + (−1.40 − 0.456i)3-s + (−0.993 + 0.113i)4-s + (−0.242 + 0.970i)5-s + (0.376 − 1.42i)6-s + 0.538i·7-s + (−0.169 − 0.985i)8-s + (0.959 + 0.696i)9-s + (−0.982 − 0.186i)10-s + (−0.228 − 0.313i)11-s + (1.44 + 0.294i)12-s + (−1.33 − 0.968i)13-s + (−0.537 + 0.0306i)14-s + (0.783 − 1.25i)15-s + (0.974 − 0.225i)16-s + (−0.450 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0646 - 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0646 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.356008 + 0.379826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.356008 + 0.379826i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 15.9i)T \) |
| 5 | \( 1 + (151. - 606. i)T \) |
| good | 3 | \( 1 + (113. + 37.0i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 1.29e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (3.33e3 + 4.59e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (3.80e4 + 2.76e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (3.76e4 + 1.15e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-2.06e5 + 6.71e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-4.80e4 - 6.61e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (1.87e5 - 5.78e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.48e6 - 4.82e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (4.30e5 + 3.12e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (5.18e5 + 3.76e5i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 9.60e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.71e6 - 1.20e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-2.66e6 + 8.20e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (8.38e6 - 1.15e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (8.09e6 - 5.87e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-7.23e6 + 2.35e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (1.33e7 + 4.34e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (4.50e7 - 3.27e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-3.19e7 - 1.03e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-6.55e7 + 2.13e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (3.62e7 - 2.63e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-9.84e6 + 3.02e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.44434228881678310186280710184, −11.63895591263221224642565650051, −10.51325880499975690093875852122, −9.254793209371452260282560652129, −7.35136780733759507241716725468, −7.12893951621253793631343232374, −5.61875064182692954002753981775, −5.13123875564973792179437706661, −3.01040344959031581509474174574, −0.51708878631192985813284061886,
0.38636042102844541116681550192, 1.71211526400010877222035602427, 3.99835212179179892352658290534, 4.76752139966875745820726137597, 5.73055776630256207362542474465, 7.59250366009817390423432764612, 9.229390234808110028030922417108, 10.06598706670980717279476149161, 11.06260117407629887563131762737, 11.98786182547060706301555900651
