| L(s) = 1 | + (151. + 87.5i)2-s + (599. − 1.03e3i)3-s + (1.12e4 + 1.94e4i)4-s − 5.45e4i·5-s + (1.81e5 − 1.05e5i)6-s + (2.44e5 − 1.41e5i)7-s + 2.50e6i·8-s + (7.73e4 + 1.34e5i)9-s + (4.77e6 − 8.27e6i)10-s + (1.27e6 + 7.36e5i)11-s + 2.69e7·12-s + (−1.71e7 − 2.92e6i)13-s + 4.94e7·14-s + (−5.67e7 − 3.27e7i)15-s + (−1.27e8 + 2.19e8i)16-s + (2.10e7 + 3.63e7i)17-s + ⋯ |
| L(s) = 1 | + (1.67 + 0.967i)2-s + (0.475 − 0.822i)3-s + (1.37 + 2.37i)4-s − 1.56i·5-s + (1.59 − 0.919i)6-s + (0.786 − 0.454i)7-s + 3.37i·8-s + (0.0485 + 0.0840i)9-s + (1.51 − 2.61i)10-s + (0.217 + 0.125i)11-s + 2.60·12-s + (−0.985 − 0.168i)13-s + 1.75·14-s + (−1.28 − 0.742i)15-s + (−1.89 + 3.27i)16-s + (0.211 + 0.365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(5.44726 + 1.01209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.44726 + 1.01209i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.71e7 + 2.92e6i)T \) |
| good | 2 | \( 1 + (-151. - 87.5i)T + (4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-599. + 1.03e3i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + 5.45e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 + (-2.44e5 + 1.41e5i)T + (4.84e10 - 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-1.27e6 - 7.36e5i)T + (1.72e13 + 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-2.10e7 - 3.63e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (8.92e7 - 5.15e7i)T + (2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (3.65e8 - 6.33e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + (1.22e9 - 2.12e9i)T + (-5.13e18 - 8.88e18i)T^{2} \) |
| 31 | \( 1 + 7.92e9iT - 2.44e19T^{2} \) |
| 37 | \( 1 + (1.34e9 + 7.77e8i)T + (1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + (3.02e10 + 1.74e10i)T + (4.62e20 + 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-4.33e9 - 7.50e9i)T + (-8.59e20 + 1.48e21i)T^{2} \) |
| 47 | \( 1 + 7.86e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 - 7.13e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + (2.51e11 - 1.45e11i)T + (5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-2.91e11 - 5.04e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.00e11 - 1.73e11i)T + (2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-3.33e10 + 1.92e10i)T + (5.82e23 - 1.00e24i)T^{2} \) |
| 73 | \( 1 - 5.43e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 1.84e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.83e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + (1.45e11 + 8.40e10i)T + (1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + (1.24e13 - 7.20e12i)T + (3.36e25 - 5.82e25i)T^{2} \) |
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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
−16.53139005670672710159092287065, −14.97203888064269681958600298130, −13.75565972691495615043304912586, −12.89567465105183177881367127466, −11.97172144902032080304104617334, −8.275344172343259790768899385368, −7.38132057593596603071326136836, −5.38038360675239050139559807366, −4.23883708389064431852307903521, −1.86354604587750408859181582874,
2.26975799262924510280209613664, 3.35705585807141385373401064918, 4.76928132131706056281892725335, 6.61868702767783483690092338227, 9.926504287835972340878154378617, 10.96339138789670754686242836479, 12.14796122677499122743812451440, 14.19745593960963523076595922433, 14.64998696304751982344008344584, 15.50808081155218370836267938809
