| L(s) = 1 | + 5.52i·3-s + 68.9i·7-s + 212.·9-s + 486.·11-s − 428. i·13-s − 1.80e3i·17-s − 1.04e3·19-s − 380.·21-s − 686. i·23-s + 2.51e3i·27-s + 1.33e3·29-s − 7.99e3·31-s + 2.68e3i·33-s + 1.97e3i·37-s + 2.36e3·39-s + ⋯ |
| L(s) = 1 | + 0.354i·3-s + 0.531i·7-s + 0.874·9-s + 1.21·11-s − 0.703i·13-s − 1.51i·17-s − 0.665·19-s − 0.188·21-s − 0.270i·23-s + 0.664i·27-s + 0.295·29-s − 1.49·31-s + 0.429i·33-s + 0.236i·37-s + 0.249·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.303986962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303986962\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 5.52iT - 243T^{2} \) |
| 7 | \( 1 - 68.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 428. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 686. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.97e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 895. iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.71e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.37e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.29e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5iT - 8.58e9T^{2} \) |
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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
−10.33360951409833008526563521672, −9.404246970849521318299539791891, −8.822181805421170441069658795519, −7.50224031263775029469969341985, −6.67406351449867704506971691763, −5.49965142119085833219657373818, −4.48008999920137193103421133022, −3.44019451854869036742521204507, −2.06075088971342133553412946218, −0.65431948913812395059529164303,
1.10089038850016783986003404680, 1.93764251902418140799259830041, 3.81543106879827016206501513245, 4.34032149058358103038009113483, 6.01114815152662507612789005336, 6.77989365943789086714528809079, 7.59991405213120838505239611597, 8.733647854361169972638458469276, 9.595256748814266546756451233302, 10.56973872947275314510707458114
