| L(s) = 1 | + (−11.4 − 11.1i)2-s + (34.8 + 84.2i)3-s + (6.38 + 255. i)4-s + (−296. + 716. i)5-s + (541. − 1.35e3i)6-s + (−3.14e3 + 3.14e3i)7-s + (2.78e3 − 3.00e3i)8-s + (−1.24e3 + 1.24e3i)9-s + (1.14e4 − 4.88e3i)10-s + (8.21e3 − 1.98e4i)11-s + (−2.13e4 + 9.46e3i)12-s + (−1.09e4 − 2.63e4i)13-s + (7.11e4 − 888. i)14-s − 7.06e4·15-s + (−6.54e4 + 3.26e3i)16-s + 7.84e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.715 − 0.698i)2-s + (0.430 + 1.04i)3-s + (0.0249 + 0.999i)4-s + (−0.474 + 1.14i)5-s + (0.417 − 1.04i)6-s + (−1.31 + 1.31i)7-s + (0.680 − 0.733i)8-s + (−0.189 + 0.189i)9-s + (1.14 − 0.488i)10-s + (0.561 − 1.35i)11-s + (−1.02 + 0.456i)12-s + (−0.381 − 0.921i)13-s + (1.85 − 0.0231i)14-s − 1.39·15-s + (−0.998 + 0.0498i)16-s + 0.939i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0533640 - 0.410225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0533640 - 0.410225i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.4 + 11.1i)T \) |
| good | 3 | \( 1 + (-34.8 - 84.2i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (296. - 716. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (3.14e3 - 3.14e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-8.21e3 + 1.98e4i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (1.09e4 + 2.63e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 7.84e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (78.8 - 32.6i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (4.02e4 + 4.02e4i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (3.24e5 - 1.34e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 1.14e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (9.71e5 - 2.34e6i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (1.60e5 - 1.60e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (1.39e6 - 3.36e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 7.12e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-3.61e6 - 1.49e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (1.12e7 + 4.65e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-6.31e6 + 2.61e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (1.11e7 + 2.69e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (2.54e7 - 2.54e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-4.40e6 + 4.40e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 3.78e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.61e7 - 1.08e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-4.51e7 - 4.51e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + 4.13e7T + 7.83e15T^{2} \) |
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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
−15.63538648797672295624380275351, −14.87842146005918462806569955488, −12.95436817137052682767603121112, −11.60389999313578794280741688010, −10.41508209583478283042358723995, −9.471198938957926136082004989411, −8.344039672927271418099572389094, −6.34978120010155090230464505036, −3.49461524805217992505281714922, −2.96537910528291085678659154673,
0.21397522168213693187298805651, 1.53437378556535281899688746368, 4.49169516972652498864250605610, 6.92361030733687223850160741925, 7.33192343812514722491022439355, 8.965003141209333232806827018633, 9.995747608605221952016984036833, 12.15896194262239718601126746394, 13.23780079951919485708262365019, 14.25114137958102507998063857589
