L(s) = 1 | + (183. − 700. i)2-s + (−2.29e4 − 2.51e4i)3-s + (−4.57e5 − 2.56e5i)4-s + 6.84e6i·5-s + (−2.18e7 + 1.14e7i)6-s − 8.10e7i·7-s + (−2.63e8 + 2.73e8i)8-s + (−1.05e8 + 1.15e9i)9-s + (4.79e9 + 1.25e9i)10-s + 1.02e10·11-s + (4.04e9 + 1.74e10i)12-s − 9.13e9·13-s + (−5.67e10 − 1.48e10i)14-s + (1.72e11 − 1.57e11i)15-s + (1.43e11 + 2.34e11i)16-s − 6.29e11i·17-s + ⋯ |
L(s) = 1 | + (0.253 − 0.967i)2-s + (−0.674 − 0.738i)3-s + (−0.871 − 0.489i)4-s + 1.56i·5-s + (−0.885 + 0.465i)6-s − 0.759i·7-s + (−0.694 + 0.719i)8-s + (−0.0911 + 0.995i)9-s + (1.51 + 0.396i)10-s + 1.31·11-s + (0.226 + 0.974i)12-s − 0.238·13-s + (−0.734 − 0.192i)14-s + (1.15 − 1.05i)15-s + (0.520 + 0.853i)16-s − 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(1.525178309\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.525178309\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-183. + 700. i)T \) |
| 3 | \( 1 + (2.29e4 + 2.51e4i)T \) |
good | 5 | \( 1 - 6.84e6iT - 1.90e13T^{2} \) |
| 7 | \( 1 + 8.10e7iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 1.02e10T + 6.11e19T^{2} \) |
| 13 | \( 1 + 9.13e9T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.29e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 - 2.01e12iT - 1.97e24T^{2} \) |
| 23 | \( 1 + 2.86e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 8.26e12iT - 6.10e27T^{2} \) |
| 31 | \( 1 + 1.75e14iT - 2.16e28T^{2} \) |
| 37 | \( 1 - 1.30e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 5.00e14iT - 4.39e30T^{2} \) |
| 43 | \( 1 + 2.29e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 - 8.03e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 2.50e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 6.40e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 4.18e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 8.57e16iT - 4.95e34T^{2} \) |
| 71 | \( 1 - 5.20e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 3.77e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 4.09e17iT - 1.13e36T^{2} \) |
| 83 | \( 1 - 1.71e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 6.97e17iT - 1.09e37T^{2} \) |
| 97 | \( 1 - 8.71e18T + 5.60e37T^{2} \) |
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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
−14.51050435449125670253846329080, −13.74852287562148514944476129482, −11.99579602492522416561085644110, −11.10542110019685259283070342860, −9.938511713992336649787968201828, −7.37307945030620058488350107936, −6.06357939260807146802833638911, −3.93581444975384391282443214241, −2.34428566887354664297221337654, −0.812669973658071531447621293473,
0.807984879168604672047683646410, 4.08090277798381087547160672431, 5.09171613569174835029096337733, 6.28947352911024065606545901350, 8.634580325577911965204685554696, 9.389809501635411577094795604195, 11.87571812135882125220172146416, 12.86814538444060508368398299946, 14.79485257363968811023415220337, 15.92208092009948568405003537579