L(s) = 1 | + (−246. − 680. i)2-s + (1.28e4 − 3.15e4i)3-s + (−4.02e5 + 3.35e5i)4-s + 2.25e5i·5-s + (−2.46e7 − 9.68e5i)6-s + 9.84e7i·7-s + (3.27e8 + 1.91e8i)8-s + (−8.32e8 − 8.11e8i)9-s + (1.53e8 − 5.55e7i)10-s + 6.04e9·11-s + (5.41e9 + 1.70e10i)12-s + 4.74e10·13-s + (6.70e10 − 2.42e10i)14-s + (7.12e9 + 2.89e9i)15-s + (4.98e10 − 2.70e11i)16-s + 8.55e11i·17-s + ⋯ |
L(s) = 1 | + (−0.340 − 0.940i)2-s + (0.376 − 0.926i)3-s + (−0.768 + 0.639i)4-s + 0.0516i·5-s + (−0.999 − 0.0392i)6-s + 0.922i·7-s + (0.863 + 0.505i)8-s + (−0.716 − 0.698i)9-s + (0.0485 − 0.0175i)10-s + 0.773·11-s + (0.302 + 0.952i)12-s + 1.24·13-s + (0.867 − 0.313i)14-s + (0.0478 + 0.0194i)15-s + (0.181 − 0.983i)16-s + 1.74i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.952i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.302 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(1.844287021\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844287021\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (246. + 680. i)T \) |
| 3 | \( 1 + (-1.28e4 + 3.15e4i)T \) |
good | 5 | \( 1 - 2.25e5iT - 1.90e13T^{2} \) |
| 7 | \( 1 - 9.84e7iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 6.04e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.74e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 8.55e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 + 3.94e11iT - 1.97e24T^{2} \) |
| 23 | \( 1 + 5.41e10T + 7.46e25T^{2} \) |
| 29 | \( 1 + 3.42e12iT - 6.10e27T^{2} \) |
| 31 | \( 1 + 2.38e14iT - 2.16e28T^{2} \) |
| 37 | \( 1 - 4.65e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 2.62e14iT - 4.39e30T^{2} \) |
| 43 | \( 1 - 2.87e13iT - 1.08e31T^{2} \) |
| 47 | \( 1 + 5.76e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.52e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 1.05e17T + 4.42e33T^{2} \) |
| 61 | \( 1 + 5.09e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.29e17iT - 4.95e34T^{2} \) |
| 71 | \( 1 - 4.53e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 8.94e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 6.91e17iT - 1.13e36T^{2} \) |
| 83 | \( 1 - 2.08e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.97e18iT - 1.09e37T^{2} \) |
| 97 | \( 1 - 9.21e17T + 5.60e37T^{2} \) |
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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
−14.86487374155000717722823793686, −13.35461183517321495834938142685, −12.33634553537274589819945151389, −11.11645722146823459492028849731, −9.116054750146157160720349170313, −8.195560449330679294050428180333, −6.16604612815841615831329739324, −3.68986762403361364378294355708, −2.19394943931718253813683280394, −1.02895650902107728308762832913,
0.893463380071435957067856761693, 3.66931746599993416237722226823, 4.99919222609535503262279963651, 6.83227617511172053752460626922, 8.467572633518650300208319971682, 9.638169773785422327463671924568, 10.96946581297834792666533487313, 13.64202048722460671818185589014, 14.47266092995944937340509424718, 15.99077889405484318075751336015