| L(s) = 1 | + (−120. + 71.3i)3-s + 275.·5-s + 3.90e3i·7-s + (9.50e3 − 1.72e4i)9-s − 8.18e4i·11-s + 1.43e5i·13-s + (−3.33e4 + 1.96e4i)15-s − 2.75e5i·17-s + 4.06e5·19-s + (−2.78e5 − 4.71e5i)21-s + 5.88e5·23-s − 1.87e6·25-s + (8.11e4 + 2.76e6i)27-s − 3.94e6·29-s + 2.49e6i·31-s + ⋯ |
| L(s) = 1 | + (−0.861 + 0.508i)3-s + 0.197·5-s + 0.614i·7-s + (0.482 − 0.875i)9-s − 1.68i·11-s + 1.39i·13-s + (−0.169 + 0.100i)15-s − 0.800i·17-s + 0.714·19-s + (−0.312 − 0.528i)21-s + 0.438·23-s − 0.961·25-s + (0.0294 + 0.999i)27-s − 1.03·29-s + 0.485i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.714i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2329267913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2329267913\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (120. - 71.3i)T \) |
| good | 5 | \( 1 - 275.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.90e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 8.18e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.43e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 2.75e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 4.06e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.88e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.94e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.49e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.06e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 6.04e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 1.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.34e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.03e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.08e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 1.02e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.21e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.12e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.61e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 3.06e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.28e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 1.06e9T + 7.60e17T^{2} \) |
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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
−10.68496476670913584373379871206, −9.398766303908269191787010292797, −8.883841083646030950927568954552, −7.26535917572604029083716315651, −6.07324145007549436040722573724, −5.45386512323064345041022601926, −4.19080286519105662815930100710, −2.96799670878121271701198890611, −1.32280088815449042195538280827, −0.06376591991409289667700215817,
1.12922019524727887169778048778, 2.21091450432466162832980056311, 3.95773267681915530728923356995, 5.13214347983152870886459523151, 6.04496411876759888135548037655, 7.33866510886357698113369231280, 7.75170779927535452088818095361, 9.559563320048259350112788356190, 10.36174811585945570429378023533, 11.15978344243318326234837690792
