| L(s) = 1 | − 22.6·2-s + (18.8 − 242. i)3-s + 512.·4-s + (−426. + 5.48e3i)6-s + 670. i·7-s − 1.15e4·8-s + (−5.83e4 − 9.12e3i)9-s − 2.33e5i·11-s + (9.64e3 − 1.24e5i)12-s − 3.07e5i·13-s − 1.51e4i·14-s + 2.62e5·16-s + 6.72e5·17-s + (1.32e6 + 2.06e5i)18-s + 1.55e6·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.0775 − 0.996i)3-s + 0.500·4-s + (−0.0548 + 0.704i)6-s + 0.0398i·7-s − 0.353·8-s + (−0.987 − 0.154i)9-s − 1.44i·11-s + (0.0387 − 0.498i)12-s − 0.828i·13-s − 0.0282i·14-s + 0.250·16-s + 0.473·17-s + (0.698 + 0.109i)18-s + 0.626·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.9329129180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9329129180\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 22.6T \) |
| 3 | \( 1 + (-18.8 + 242. i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 670. iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.33e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.07e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 - 6.72e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.55e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 5.57e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 2.97e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.09e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 8.56e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 3.59e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 3.66e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 3.28e7T + 5.25e16T^{2} \) |
| 53 | \( 1 + 4.59e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 4.88e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 6.12e7T + 7.13e17T^{2} \) |
| 67 | \( 1 + 6.70e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.23e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.08e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 1.86e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 1.09e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 5.19e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.07e10iT - 7.37e19T^{2} \) |
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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
−10.57596540499924111605082313752, −9.330057435640598460191562345860, −8.183694796573246844881843276332, −7.73178818196458479851073832176, −6.31748778654170071500278733496, −5.61171239955262146575405816556, −3.42446971051215239225600529119, −2.38692124945801935214171434196, −0.988602187253098440295486836816, −0.30201616462484960846975386877,
1.44982910169287158763740065236, 2.75333119558039016847702495297, 4.11170392903757222581133334658, 5.16140772487430150389119930169, 6.59589449175288498047499661534, 7.73970606344221288838860321862, 8.881790798236681202262144099652, 9.776276183101144377362020869993, 10.32043969183038222716738511269, 11.55695950157271927552775589416
