| L(s) = 1 | + (−6.67e4 + 7.75e4i)3-s + 3.35e6i·5-s + 5.38e8i·7-s + (−1.55e9 − 1.03e10i)9-s − 1.10e10·11-s − 1.02e11·13-s + (−2.60e11 − 2.24e11i)15-s − 7.33e12i·17-s + 2.47e13i·19-s + (−4.17e13 − 3.59e13i)21-s + 1.19e13·23-s + 4.65e14·25-s + (9.05e14 + 5.69e14i)27-s + 1.07e15i·29-s − 2.35e15i·31-s + ⋯ |
| L(s) = 1 | + (−0.652 + 0.757i)3-s + 0.153i·5-s + 0.720i·7-s + (−0.148 − 0.988i)9-s − 0.128·11-s − 0.206·13-s + (−0.116 − 0.100i)15-s − 0.883i·17-s + 0.925i·19-s + (−0.546 − 0.470i)21-s + 0.0602·23-s + 0.976·25-s + (0.846 + 0.532i)27-s + 0.472i·29-s − 0.515i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.398404851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398404851\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.67e4 - 7.75e4i)T \) |
| good | 5 | \( 1 - 3.35e6iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 5.38e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 1.10e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 1.02e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.33e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 - 2.47e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 1.19e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.07e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 2.35e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 9.92e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 5.00e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 - 2.92e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 6.06e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.40e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 2.94e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 2.60e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.51e18iT - 2.22e38T^{2} \) |
| 71 | \( 1 - 2.03e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.87e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 3.90e19iT - 7.08e39T^{2} \) |
| 83 | \( 1 + 2.01e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.22e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 + 4.24e20T + 5.27e41T^{2} \) |
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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
−11.75201902052793363557812499898, −10.69065527609132304393459245627, −9.662601657766060074658003063636, −8.647385335456809902166279321789, −7.05416247714538501946028029310, −5.79974660842564491332072275737, −4.96575846274951481440100003466, −3.66650915606316379764634186654, −2.44435600366628382157966708376, −0.801348026078628307138763371454,
0.43525063236696699270705752673, 1.30505589998885880841678078947, 2.60875973009896567837085047772, 4.23996386372743330289787988930, 5.38527910845424240311029380000, 6.61934518124092454743001579847, 7.47758462270215337614582672883, 8.688296001796466688774378662791, 10.30409709621046254533755161960, 11.13062890826643083661149360243
