| L(s) = 1 | + (16 − 27.7i)2-s + (69.1 + 119. i)3-s + (−511. − 886. i)4-s + (274. − 475. i)5-s + 4.42e3·6-s + (−2.54e4 − 3.64e4i)7-s − 3.27e4·8-s + (7.89e4 − 1.36e5i)9-s + (−8.78e3 − 1.52e4i)10-s + (−1.23e5 − 2.14e5i)11-s + (7.08e4 − 1.22e5i)12-s − 1.83e6·13-s + (−1.41e6 + 1.22e5i)14-s + 7.59e4·15-s + (−5.24e5 + 9.08e5i)16-s + (−3.08e6 − 5.34e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.164 + 0.284i)3-s + (−0.249 − 0.433i)4-s + (0.0392 − 0.0680i)5-s + 0.232·6-s + (−0.572 − 0.819i)7-s − 0.353·8-s + (0.445 − 0.772i)9-s + (−0.0277 − 0.0481i)10-s + (−0.232 − 0.401i)11-s + (0.0821 − 0.142i)12-s − 1.37·13-s + (−0.704 + 0.0606i)14-s + 0.0258·15-s + (−0.125 + 0.216i)16-s + (−0.526 − 0.912i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.495626 - 1.41723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495626 - 1.41723i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 + 27.7i)T \) |
| 7 | \( 1 + (2.54e4 + 3.64e4i)T \) |
| good | 3 | \( 1 + (-69.1 - 119. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-274. + 475. i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.23e5 + 2.14e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.83e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (3.08e6 + 5.34e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.41e5 + 1.45e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-8.79e6 + 1.52e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.50e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-5.90e7 - 1.02e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.77e8 + 4.80e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 4.42e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.11e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-3.64e7 + 6.31e7i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (5.60e8 + 9.71e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-4.44e9 - 7.70e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (4.28e8 - 7.41e8i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.95e9 + 3.39e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.40e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.22e10 + 2.12e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.75e10 + 3.04e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 6.12e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.76e10 - 3.05e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.31e11T + 7.15e21T^{2} \) |
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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
−16.24719722981417723644809008306, −14.78587655403221320252555947379, −13.43798497931319538396415820548, −12.16171018592050140644198848855, −10.45183290897099640750122171845, −9.309169481472270580676297316384, −6.92028360030846654754261650210, −4.65757345000322346618800266234, −3.01337058184945248721127549766, −0.59310987018229284807386746760,
2.48000016799933002580070895472, 4.85666899907774975184667146611, 6.64943066780886379378273731135, 8.171390266183584925889307877470, 9.965744979814440163484810013192, 12.21324878060225996949618623310, 13.25467259400661917896973696365, 14.79171453158729007402749755231, 15.88152647654551616672627257194, 17.26352515237617088884122843260
