L(s) = 1 | + (519. − 504. i)2-s + (1.97e4 + 2.77e4i)3-s + (1.55e4 − 5.24e5i)4-s − 1.34e6i·5-s + (2.42e7 + 4.44e6i)6-s + 1.72e8i·7-s + (−2.56e8 − 2.80e8i)8-s + (−3.79e8 + 1.09e9i)9-s + (−6.78e8 − 6.98e8i)10-s + 1.32e10·11-s + (1.48e10 − 9.93e9i)12-s − 2.84e10·13-s + (8.70e10 + 8.96e10i)14-s + (3.73e10 − 2.66e10i)15-s + (−2.74e11 − 1.62e10i)16-s + 1.96e11i·17-s + ⋯ |
L(s) = 1 | + (0.717 − 0.696i)2-s + (0.580 + 0.814i)3-s + (0.0295 − 0.999i)4-s − 0.307i·5-s + (0.983 + 0.180i)6-s + 1.61i·7-s + (−0.675 − 0.737i)8-s + (−0.326 + 0.945i)9-s + (−0.214 − 0.220i)10-s + 1.70·11-s + (0.831 − 0.555i)12-s − 0.745·13-s + (1.12 + 1.15i)14-s + (0.250 − 0.178i)15-s + (−0.998 − 0.0591i)16-s + 0.401i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(3.508596092\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.508596092\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-519. + 504. i)T \) |
| 3 | \( 1 + (-1.97e4 - 2.77e4i)T \) |
good | 5 | \( 1 + 1.34e6iT - 1.90e13T^{2} \) |
| 7 | \( 1 - 1.72e8iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 1.32e10T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.84e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 1.96e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 - 2.26e12iT - 1.97e24T^{2} \) |
| 23 | \( 1 - 9.34e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.66e13iT - 6.10e27T^{2} \) |
| 31 | \( 1 + 4.90e13iT - 2.16e28T^{2} \) |
| 37 | \( 1 - 9.66e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.16e15iT - 4.39e30T^{2} \) |
| 43 | \( 1 - 3.53e14iT - 1.08e31T^{2} \) |
| 47 | \( 1 + 2.94e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 9.59e15iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 1.43e15T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.36e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.13e17iT - 4.95e34T^{2} \) |
| 71 | \( 1 + 4.79e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 3.71e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 5.54e17iT - 1.13e36T^{2} \) |
| 83 | \( 1 + 1.99e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.47e18iT - 1.09e37T^{2} \) |
| 97 | \( 1 + 5.09e18T + 5.60e37T^{2} \) |
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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
−15.00087838347816478718769770042, −14.52341832075352929996233125278, −12.61282059987894978278562786856, −11.53313456031944535356342275951, −9.725772377839648738969274685527, −8.804815358372884112292356461169, −5.92113565069519960003354959781, −4.54082284188372693689218475534, −3.10226650012275257202205042275, −1.71961640215708260234831617054,
0.887425879770882160475334759139, 2.99388387121091578368074904860, 4.36916096355967958678291039862, 6.80584775157536947262843187830, 7.20431787944548775488078034241, 9.055122556559390637799471788260, 11.44582186346067906064456404416, 12.99298526509404278970614879413, 14.04374708740957631484146070244, 14.80831428179404719447209815912