| L(s) = 1 | + (3.54 − 13.2i)2-s + (29.1 + 50.4i)3-s + (58.8 + 33.9i)4-s + (367. + 367. i)5-s + (771. − 206. i)6-s + (60.5 + 225. i)7-s + (3.14e3 − 3.14e3i)8-s + (1.58e3 − 2.74e3i)9-s + (6.16e3 − 3.56e3i)10-s + (1.99e3 + 535. i)11-s + 3.95e3i·12-s + (−6.05e3 + 2.79e4i)13-s + 3.20e3·14-s + (−7.83e3 + 2.92e4i)15-s + (−2.17e4 − 3.76e4i)16-s + (−6.86e4 − 3.96e4i)17-s + ⋯ |
| L(s) = 1 | + (0.221 − 0.827i)2-s + (0.359 + 0.623i)3-s + (0.229 + 0.132i)4-s + (0.587 + 0.587i)5-s + (0.595 − 0.159i)6-s + (0.0252 + 0.0941i)7-s + (0.766 − 0.766i)8-s + (0.241 − 0.417i)9-s + (0.616 − 0.356i)10-s + (0.136 + 0.0365i)11-s + 0.190i·12-s + (−0.211 + 0.977i)13-s + 0.0835·14-s + (−0.154 + 0.577i)15-s + (−0.332 − 0.575i)16-s + (−0.821 − 0.474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.29835 - 0.201555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29835 - 0.201555i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (6.05e3 - 2.79e4i)T \) |
| good | 2 | \( 1 + (-3.54 + 13.2i)T + (-221. - 128i)T^{2} \) |
| 3 | \( 1 + (-29.1 - 50.4i)T + (-3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-367. - 367. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + (-60.5 - 225. i)T + (-4.99e6 + 2.88e6i)T^{2} \) |
| 11 | \( 1 + (-1.99e3 - 535. i)T + (1.85e8 + 1.07e8i)T^{2} \) |
| 17 | \( 1 + (6.86e4 + 3.96e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.65e4 - 1.51e4i)T + (1.47e10 - 8.49e9i)T^{2} \) |
| 23 | \( 1 + (7.85e4 - 4.53e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (9.60e4 + 1.66e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (9.31e5 + 9.31e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (2.56e6 + 6.88e5i)T + (3.04e12 + 1.75e12i)T^{2} \) |
| 41 | \( 1 + (1.02e5 - 3.82e5i)T + (-6.91e12 - 3.99e12i)T^{2} \) |
| 43 | \( 1 + (6.94e5 + 4.00e5i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (3.58e6 - 3.58e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 - 1.20e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.36e6 - 1.62e7i)T + (-1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (5.20e6 - 9.01e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-4.37e6 + 1.63e7i)T + (-3.51e14 - 2.03e14i)T^{2} \) |
| 71 | \( 1 + (1.91e7 - 5.14e6i)T + (5.59e14 - 3.22e14i)T^{2} \) |
| 73 | \( 1 + (-2.09e7 + 2.09e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 6.65e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-5.29e7 - 5.29e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + (6.21e7 + 1.66e7i)T + (3.40e15 + 1.96e15i)T^{2} \) |
| 97 | \( 1 + (1.19e8 - 3.20e7i)T + (6.78e15 - 3.91e15i)T^{2} \) |
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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
−18.17552533311430803408800182782, −16.46401626742012628642714273074, −15.06379849067498648867350939032, −13.65013181111221187722807124969, −12.00925370696325775901401530937, −10.60059405581539921696695698449, −9.354977955087996307449087697888, −6.80016319774104639255494928284, −3.99961793206053340086054703388, −2.24098697611729680511548591441,
1.81934166071623873898266934442, 5.26503170451840423337636262423, 6.95019345724222209098475116910, 8.423909511554372119152728681567, 10.59270836189415793152659799604, 12.79752502188065130198435206384, 13.85086364384068706058716198661, 15.24241816796536993345839028807, 16.58902264699207458721654050502, 17.74328449095016679016877286464
