| L(s) = 1 | + (340. + 965. i)2-s + (−8.16e5 + 6.57e5i)4-s − 4.86e6i·5-s − 1.26e8i·7-s + (−9.13e8 − 5.64e8i)8-s + (4.69e9 − 1.65e9i)10-s − 1.51e10·11-s + 1.95e11i·13-s + (1.21e11 − 4.29e10i)14-s + (2.34e11 − 1.07e12i)16-s − 3.56e12·17-s + 3.48e12·19-s + (3.19e12 + 3.97e12i)20-s + (−5.15e12 − 1.46e13i)22-s + 1.17e13i·23-s + ⋯ |
| L(s) = 1 | + (0.332 + 0.943i)2-s + (−0.778 + 0.627i)4-s − 0.498i·5-s − 0.446i·7-s + (−0.850 − 0.525i)8-s + (0.469 − 0.165i)10-s − 0.583·11-s + 1.41i·13-s + (0.421 − 0.148i)14-s + (0.213 − 0.977i)16-s − 1.76·17-s + 0.568·19-s + (0.312 + 0.388i)20-s + (−0.194 − 0.550i)22-s + 0.284i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.7654040356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7654040356\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-340. - 965. i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.86e6iT - 9.53e13T^{2} \) |
| 7 | \( 1 + 1.26e8iT - 7.97e16T^{2} \) |
| 11 | \( 1 + 1.51e10T + 6.72e20T^{2} \) |
| 13 | \( 1 - 1.95e11iT - 1.90e22T^{2} \) |
| 17 | \( 1 + 3.56e12T + 4.06e24T^{2} \) |
| 19 | \( 1 - 3.48e12T + 3.75e25T^{2} \) |
| 23 | \( 1 - 1.17e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 7.56e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 1.11e15iT - 6.71e29T^{2} \) |
| 37 | \( 1 + 7.17e14iT - 2.31e31T^{2} \) |
| 41 | \( 1 + 3.23e15T + 1.80e32T^{2} \) |
| 43 | \( 1 + 1.36e16T + 4.67e32T^{2} \) |
| 47 | \( 1 - 4.28e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 3.34e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + 2.74e17T + 2.61e35T^{2} \) |
| 61 | \( 1 - 1.49e17iT - 5.08e35T^{2} \) |
| 67 | \( 1 + 2.92e18T + 3.32e36T^{2} \) |
| 71 | \( 1 + 3.51e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 5.45e17T + 1.84e37T^{2} \) |
| 79 | \( 1 + 1.64e19iT - 8.96e37T^{2} \) |
| 83 | \( 1 - 1.30e19T + 2.40e38T^{2} \) |
| 89 | \( 1 - 2.43e19T + 9.72e38T^{2} \) |
| 97 | \( 1 + 7.85e18T + 5.43e39T^{2} \) |
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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
−10.75240276014766051355613528611, −9.210255246427311067175378089116, −8.622238264107503278817704501221, −7.22488412587833919185891483882, −6.57212113307780449928910907010, −5.08019941004999472690710482884, −4.47048736186471220195555975224, −3.21069601337247221449448321327, −1.60652008887949036397599317286, −0.16583172182367863249182895938,
0.74893430565187649141002985479, 2.32025481954450515926286890514, 2.81389613294035523067551141526, 4.14031152330583582869256518926, 5.28006838041834689132193514656, 6.27403084852164483774310070787, 7.87545368277992459107898635033, 9.010458000486399951209411051603, 10.17271808668377800558839392377, 10.93113821566306327427941725682
