| L(s) = 1 | + (51.7 − 37.6i)2-s + (346. + 1.06e3i)3-s + (1.26e3 − 3.89e3i)4-s + (−2.49e4 − 1.81e4i)5-s + (5.80e4 + 4.21e4i)6-s + (−7.97e3 + 2.45e4i)7-s + (−8.10e4 − 2.49e5i)8-s + (2.72e5 − 1.97e5i)9-s − 1.97e6·10-s + (−2.54e6 + 5.29e6i)11-s + 4.59e6·12-s + (−2.21e7 + 1.61e7i)13-s + (5.10e5 + 1.57e6i)14-s + (1.06e7 − 3.28e7i)15-s + (−1.35e7 − 9.86e6i)16-s + (−4.73e7 − 3.44e7i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.274 + 0.844i)3-s + (0.154 − 0.475i)4-s + (−0.713 − 0.518i)5-s + (0.508 + 0.369i)6-s + (−0.0256 + 0.0788i)7-s + (−0.109 − 0.336i)8-s + (0.170 − 0.124i)9-s − 0.623·10-s + (−0.433 + 0.900i)11-s + 0.444·12-s + (−1.27 + 0.925i)13-s + (0.0181 + 0.0557i)14-s + (0.242 − 0.744i)15-s + (−0.202 − 0.146i)16-s + (−0.476 − 0.346i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.04322693121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04322693121\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + (2.54e6 - 5.29e6i)T \) |
| good | 3 | \( 1 + (-346. - 1.06e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (2.49e4 + 1.81e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (7.97e3 - 2.45e4i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (2.21e7 - 1.61e7i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (4.73e7 + 3.44e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (8.90e7 + 2.74e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 + 1.25e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-6.92e7 + 2.13e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (6.90e9 - 5.01e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-1.06e9 + 3.27e9i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-6.01e9 - 1.85e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 1.14e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (8.18e9 + 2.51e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-1.55e11 + 1.12e11i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-3.42e10 + 1.05e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-2.41e11 - 1.75e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 1.00e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-6.20e11 - 4.50e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (8.28e10 - 2.55e11i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-2.83e11 + 2.05e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (2.99e12 + 2.17e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 - 1.42e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-1.16e13 + 8.45e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−15.43147247943367207664552979989, −14.40268202435840808223722301352, −12.73506828988425659965595701963, −11.76565526233715622797748945853, −10.18104872235287629325749014304, −9.081780988372171939060691241322, −7.13907132546156869508145429058, −4.81354634754920157977693284198, −4.12217597986159986979370930710, −2.27160311725749096150607039481,
0.01073589414195207135285594375, 2.28475868930771200294668138986, 3.84285286932814287860181727866, 5.82008942215334266950309988385, 7.41350246568790156478610009174, 8.060188560213461346087067254209, 10.47635137210508073741471480824, 12.06223574771026538268217936895, 13.06274893742576300264226852989, 14.27283455664909648781739982896
