| L(s) = 1 | + 22.6i·2-s + (−242. + 18.8i)3-s − 512.·4-s + (−426. − 5.48e3i)6-s − 670.·7-s − 1.15e4i·8-s + (5.83e4 − 9.12e3i)9-s + 2.33e5i·11-s + (1.24e5 − 9.64e3i)12-s − 3.07e5·13-s − 1.51e4i·14-s + 2.62e5·16-s − 6.72e5i·17-s + (2.06e5 + 1.32e6i)18-s − 1.55e6·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.996 + 0.0775i)3-s − 0.500·4-s + (−0.0548 − 0.704i)6-s − 0.0398·7-s − 0.353i·8-s + (0.987 − 0.154i)9-s + 1.44i·11-s + (0.498 − 0.0387i)12-s − 0.828·13-s − 0.0282i·14-s + 0.250·16-s − 0.473i·17-s + (0.109 + 0.698i)18-s − 0.626·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.9791418229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9791418229\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 22.6iT \) |
| 3 | \( 1 + (242. - 18.8i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 670.T + 2.82e8T^{2} \) |
| 11 | \( 1 - 2.33e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.07e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 6.72e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.55e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 5.57e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 2.97e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.09e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 8.56e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 3.59e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 3.66e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 3.28e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 4.59e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 4.88e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 6.12e7T + 7.13e17T^{2} \) |
| 67 | \( 1 - 6.70e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.23e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.08e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 1.86e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 1.09e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 5.19e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.07e10T + 7.37e19T^{2} \) |
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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
−11.49646362514076582608794065651, −10.10885866490591533213353263103, −9.612881871682142893139225633210, −7.990129302977186820070945912151, −7.00209119703822747643041474916, −6.21098392912219554133165199815, −4.88701230192042489337729329011, −4.33151112005138653925475220774, −2.24814984826177286342936833796, −0.62522172422253447069246005895,
0.43318599030416082252825453761, 1.47934878581305244975120821724, 2.98898371408951049657578678796, 4.29051114786963006138814285027, 5.42351706544423124354609532665, 6.37474396967391076727199709168, 7.75591532803140889154237088565, 9.003539121504625428630280068312, 10.17427529040560204773922608366, 10.95852567661519848950643904527
