| L(s) = 1 | + (−51.7 + 37.6i)2-s + (41.6 + 128. i)3-s + (1.26e3 − 3.89e3i)4-s + (2.57e3 + 1.86e3i)5-s + (−6.97e3 − 5.06e3i)6-s + (−2.70e4 + 8.33e4i)7-s + (8.10e4 + 2.49e5i)8-s + (1.27e6 − 9.26e5i)9-s − 2.03e5·10-s + (−5.72e6 + 1.30e6i)11-s + 5.51e5·12-s + (−1.02e7 + 7.46e6i)13-s + (−1.73e6 − 5.33e6i)14-s + (−1.32e5 + 4.07e5i)15-s + (−1.35e7 − 9.86e6i)16-s + (−3.54e7 − 2.57e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.0329 + 0.101i)3-s + (0.154 − 0.475i)4-s + (0.0736 + 0.0535i)5-s + (−0.0610 − 0.0443i)6-s + (−0.0869 + 0.267i)7-s + (0.109 + 0.336i)8-s + (0.799 − 0.581i)9-s − 0.0643·10-s + (−0.975 + 0.221i)11-s + 0.0533·12-s + (−0.590 + 0.428i)13-s + (−0.0615 − 0.189i)14-s + (−0.00300 + 0.00923i)15-s + (−0.202 − 0.146i)16-s + (−0.356 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9906992860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9906992860\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (51.7 - 37.6i)T \) |
| 11 | \( 1 + (5.72e6 - 1.30e6i)T \) |
| good | 3 | \( 1 + (-41.6 - 128. i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-2.57e3 - 1.86e3i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (2.70e4 - 8.33e4i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (1.02e7 - 7.46e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (3.54e7 + 2.57e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (1.24e7 + 3.82e7i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 6.03e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-1.42e9 + 4.37e9i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-6.94e9 + 5.04e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-3.43e9 + 1.05e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (4.97e9 + 1.53e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 + 2.23e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.99e10 + 6.13e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (8.73e10 - 6.34e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (1.09e10 - 3.36e10i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (4.00e11 + 2.91e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 - 8.63e10T + 5.48e23T^{2} \) |
| 71 | \( 1 + (8.06e11 + 5.86e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-3.21e11 + 9.89e11i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-2.62e12 + 1.90e12i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (1.54e12 + 1.12e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 - 3.91e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.27e12 - 9.25e11i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−15.10649083941152809293575540144, −13.48670395469771867351281561321, −12.01243285473756165043218803663, −10.33341553074557570611958840461, −9.306849264674594558052239116810, −7.73567469703170322268221815218, −6.39851606451544632622255684293, −4.63275023467291246517071663265, −2.36912999104031743693616754956, −0.43501745128604333925591712492,
1.29650213981203910283843427835, 2.91772224755174418203476416386, 4.90046816306498866366508879504, 7.06683413208378368033562675618, 8.291072179979537640832547923005, 9.954632753036385353722763537418, 10.86626823003050903695326145899, 12.55836579355879855625575141263, 13.50131142742507981211805446447, 15.34325907277774223468338945844
