| L(s) = 1 | + (−1.73 − 2.38i)2-s + (−394. − 1.21e3i)3-s + (1.26e3 − 3.88e3i)4-s + (−1.22e4 − 8.93e3i)5-s + (−2.21e3 + 3.04e3i)6-s + (1.06e5 + 3.45e4i)7-s + (−2.29e4 + 7.45e3i)8-s + (−8.88e5 + 6.45e5i)9-s + 4.47e4i·10-s + (1.58e6 − 7.86e5i)11-s − 5.21e6·12-s + (−1.15e6 − 1.59e6i)13-s + (−1.01e5 − 3.13e5i)14-s + (−5.99e6 + 1.84e7i)15-s + (−1.34e7 − 9.79e6i)16-s + (−1.87e7 + 2.57e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.0270 − 0.0372i)2-s + (−0.541 − 1.66i)3-s + (0.308 − 0.949i)4-s + (−0.786 − 0.571i)5-s + (−0.0473 + 0.0652i)6-s + (0.903 + 0.293i)7-s + (−0.0875 + 0.0284i)8-s + (−1.67 + 1.21i)9-s + 0.0447i·10-s + (0.896 − 0.443i)11-s − 1.74·12-s + (−0.239 − 0.329i)13-s + (−0.0135 − 0.0415i)14-s + (−0.526 + 1.61i)15-s + (−0.803 − 0.584i)16-s + (−0.775 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.233916 + 1.11660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233916 + 1.11660i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.58e6 + 7.86e5i)T \) |
| good | 2 | \( 1 + (1.73 + 2.38i)T + (-1.26e3 + 3.89e3i)T^{2} \) |
| 3 | \( 1 + (394. + 1.21e3i)T + (-4.29e5 + 3.12e5i)T^{2} \) |
| 5 | \( 1 + (1.22e4 + 8.93e3i)T + (7.54e7 + 2.32e8i)T^{2} \) |
| 7 | \( 1 + (-1.06e5 - 3.45e4i)T + (1.11e10 + 8.13e9i)T^{2} \) |
| 13 | \( 1 + (1.15e6 + 1.59e6i)T + (-7.19e12 + 2.21e13i)T^{2} \) |
| 17 | \( 1 + (1.87e7 - 2.57e7i)T + (-1.80e14 - 5.54e14i)T^{2} \) |
| 19 | \( 1 + (-6.37e7 + 2.07e7i)T + (1.79e15 - 1.30e15i)T^{2} \) |
| 23 | \( 1 - 1.27e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (4.42e8 + 1.43e8i)T + (2.86e17 + 2.07e17i)T^{2} \) |
| 31 | \( 1 + (3.20e7 - 2.33e7i)T + (2.43e17 - 7.49e17i)T^{2} \) |
| 37 | \( 1 + (8.67e8 - 2.67e9i)T + (-5.32e18 - 3.86e18i)T^{2} \) |
| 41 | \( 1 + (4.29e9 - 1.39e9i)T + (1.82e19 - 1.32e19i)T^{2} \) |
| 43 | \( 1 + 1.20e10iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (1.58e9 + 4.86e9i)T + (-9.40e19 + 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-1.92e10 + 1.39e10i)T + (1.51e20 - 4.67e20i)T^{2} \) |
| 59 | \( 1 + (1.37e10 - 4.24e10i)T + (-1.43e21 - 1.04e21i)T^{2} \) |
| 61 | \( 1 + (-2.45e10 + 3.37e10i)T + (-8.20e20 - 2.52e21i)T^{2} \) |
| 67 | \( 1 - 3.93e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (1.13e11 + 8.24e10i)T + (5.07e21 + 1.56e22i)T^{2} \) |
| 73 | \( 1 + (-1.38e11 - 4.51e10i)T + (1.85e22 + 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-1.85e11 - 2.54e11i)T + (-1.82e22 + 5.61e22i)T^{2} \) |
| 83 | \( 1 + (1.20e11 - 1.65e11i)T + (-3.30e22 - 1.01e23i)T^{2} \) |
| 89 | \( 1 + 3.43e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (1.04e11 - 7.62e10i)T + (2.14e23 - 6.59e23i)T^{2} \) |
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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
−17.02892642613384790409993133209, −15.15548654608065429070953070678, −13.63375352024227978432181237211, −12.00466303874781011440368599235, −11.26290362204380682717610083833, −8.450719015777396998746628954905, −6.90331197662565923310437789943, −5.35318870213478082942079944288, −1.69875476994015376442614585034, −0.60509398321663153410383249644,
3.46850288034603281428940993878, 4.65859205092023160939356579433, 7.29539410185381702515442179648, 9.259692154087417807677873570727, 11.17609942468605703759066108857, 11.67684015305665225840731551660, 14.52252784584898340213514973700, 15.67087481436563250751797065140, 16.68670985209533104960739857811, 17.79099009609557387334645024533
