| L(s) = 1 | + 9·3-s − 86.8i·5-s − 98.7i·7-s + 81·9-s − 610. i·11-s + (592. − 143. i)13-s − 781. i·15-s − 1.14e3·17-s − 2.26e3i·19-s − 888. i·21-s − 433.·23-s − 4.41e3·25-s + 729·27-s + 7.66e3·29-s − 7.36e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.55i·5-s − 0.761i·7-s + 0.333·9-s − 1.52i·11-s + (0.971 − 0.234i)13-s − 0.897i·15-s − 0.963·17-s − 1.44i·19-s − 0.439i·21-s − 0.170·23-s − 1.41·25-s + 0.192·27-s + 1.69·29-s − 1.37i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.234i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.690019049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690019049\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + (-592. + 143. i)T \) |
| good | 5 | \( 1 + 86.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 98.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 610. iT - 1.61e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.26e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 433.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.36e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.05e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 3.69e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 6.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.74e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.19e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.56e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.38e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 3.77e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.14e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.90e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 4.96e3iT - 8.58e9T^{2} \) |
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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
−9.029098946886832169529376897368, −8.678219701210480503264542930186, −8.024821882266853532768251249942, −6.75869678639090112478506835928, −5.73156100919380872246432907298, −4.61017018053627682220870614476, −3.91239337289357224795388079396, −2.65351100516922038907593373695, −1.04640521067619181113254337760, −0.60399422954393106045835562205,
1.74006164358501563498803382170, 2.50460952834648100192678411711, 3.49548563156376494989870306763, 4.51850985886635876298616475890, 6.01070420502578545403821063739, 6.72659945489127875809895469565, 7.51174484146521779400399471214, 8.507688236771129014688361174740, 9.387018642100795803142021835039, 10.34502308784576475178183050912
