| L(s) = 1 | − 10.4i·3-s − 66.0·7-s + 134.·9-s − 141. i·11-s + 246. i·13-s + 297.·17-s + 174. i·19-s + 688. i·21-s − 1.57e3·23-s − 3.93e3i·27-s − 922. i·29-s + 6.19e3·31-s − 1.48e3·33-s + 1.39e4i·37-s + 2.57e3·39-s + ⋯ |
| L(s) = 1 | − 0.669i·3-s − 0.509·7-s + 0.551·9-s − 0.353i·11-s + 0.405i·13-s + 0.249·17-s + 0.110i·19-s + 0.340i·21-s − 0.621·23-s − 1.03i·27-s − 0.203i·29-s + 1.15·31-s − 0.236·33-s + 1.67i·37-s + 0.271·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.061404021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061404021\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.4iT - 243T^{2} \) |
| 7 | \( 1 + 66.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 141. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 246. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 297.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 174. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 922. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 3.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.62e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.29e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.18e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.86e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.67e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e4T + 8.58e9T^{2} \) |
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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
−9.685640067457867980143483927578, −8.391260525561573212425340425987, −7.81867912056846020891718741817, −6.65646778595259152699069746984, −6.33719263950396371190328081128, −5.01503260745124788954807165700, −3.96970796570595400676776765688, −2.86827947571865473398636421542, −1.70531085800221983416647956176, −0.71253368852011956101927937950,
0.61569480339858706485956968350, 2.00904764205824002901070685715, 3.26741638218319423441235663517, 4.11516027679407723827275585769, 5.01776695996754846788992161120, 6.01226987499762360570698170975, 7.00608283402982099693872866862, 7.82381031681308660399881988319, 8.918444307312892919303298818817, 9.679241686023298532512793844410
