| L(s) = 1 | + (14.6 − 6.47i)2-s + (−23.9 − 7.77i)3-s + (172. − 189. i)4-s + (−439. − 444. i)5-s + (−400. + 41.2i)6-s + 3.46e3i·7-s + (1.29e3 − 3.88e3i)8-s + (−4.79e3 − 3.48e3i)9-s + (−9.30e3 − 3.65e3i)10-s + (−1.19e3 − 1.64e3i)11-s + (−5.58e3 + 3.19e3i)12-s + (8.83e3 + 6.41e3i)13-s + (2.24e4 + 5.06e4i)14-s + (7.05e3 + 1.40e4i)15-s + (−6.29e3 − 6.52e4i)16-s + (3.16e4 + 9.75e4i)17-s + ⋯ |
| L(s) = 1 | + (0.914 − 0.404i)2-s + (−0.295 − 0.0959i)3-s + (0.672 − 0.740i)4-s + (−0.702 − 0.711i)5-s + (−0.308 + 0.0317i)6-s + 1.44i·7-s + (0.315 − 0.949i)8-s + (−0.731 − 0.531i)9-s + (−0.930 − 0.365i)10-s + (−0.0814 − 0.112i)11-s + (−0.269 + 0.154i)12-s + (0.309 + 0.224i)13-s + (0.584 + 1.31i)14-s + (0.139 + 0.277i)15-s + (−0.0960 − 0.995i)16-s + (0.379 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.06276 + 0.775890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06276 + 0.775890i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-14.6 + 6.47i)T \) |
| 5 | \( 1 + (439. + 444. i)T \) |
| good | 3 | \( 1 + (23.9 + 7.77i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 3.46e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.19e3 + 1.64e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-8.83e3 - 6.41e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-3.16e4 - 9.75e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.62e5 - 5.28e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-5.68e4 - 7.82e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (9.87e4 - 3.03e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-8.03e5 + 2.60e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.01e6 - 7.40e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (2.18e6 + 1.58e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.79e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.45e6 - 1.12e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (1.99e6 - 6.14e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (9.22e6 - 1.26e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.63e6 + 1.19e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (7.74e6 - 2.51e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (6.69e6 + 2.17e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.94e7 - 1.41e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (4.62e7 + 1.50e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-8.40e7 + 2.73e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-1.73e7 + 1.25e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (4.13e6 - 1.27e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.29175861104992721863976944390, −11.83790183414577584689807280974, −10.79167292126769793136324727348, −9.162363751554820241768153027465, −8.212524317166834134858096931892, −6.28359955283519717103951378300, −5.58773075406231214455445805561, −4.26172408824205311264923260181, −2.93970140291988423251116578190, −1.41085925655861522811792452978,
0.27871351655878376477254078521, 2.66495957598768266450140975346, 3.87775725950780577869520380016, 4.91066856654362693261711948672, 6.44747612757503871062088476177, 7.33184356301064688494397648794, 8.262069898619957768720097446895, 10.45947838167496656753691040037, 11.07194011106761022504208886458, 12.01178945595069814319457725884
