L(s) = 1 | − 14.3·5-s + 532.·7-s + 1.84e3·11-s − 1.00e3i·13-s − 3.71e3i·17-s − 7.86e3i·19-s − 1.95e4i·23-s − 1.54e4·25-s − 4.25e4·29-s + 3.91e4·31-s − 7.65e3·35-s − 6.17e4i·37-s + 1.33e4i·41-s + 5.79e4i·43-s − 5.19e4i·47-s + ⋯ |
L(s) = 1 | − 0.115·5-s + 1.55·7-s + 1.38·11-s − 0.457i·13-s − 0.756i·17-s − 1.14i·19-s − 1.60i·23-s − 0.986·25-s − 1.74·29-s + 1.31·31-s − 0.178·35-s − 1.22i·37-s + 0.193i·41-s + 0.728i·43-s − 0.500i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.415625484\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.415625484\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 14.3T + 1.56e4T^{2} \) |
| 7 | \( 1 - 532.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.84e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.00e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 3.71e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.86e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.95e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.25e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.91e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.17e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.33e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.79e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 5.19e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 5.29e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 9.58e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.48e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.70e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.77e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.81e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.58e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.58e4T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.09e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.01e6T + 8.32e11T^{2} \) |
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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
−9.363889680102565789067585099803, −8.659987622370594183186414014117, −7.77066558541804301039696909691, −6.93999185575583878368438482957, −5.81849861898771753232941039914, −4.73890955725630651252105509046, −4.08338610808573090297217982652, −2.60193188176512494966197625569, −1.51180795437735053140645701523, −0.46947461899389661693035308910,
1.43470163946398115371192750314, 1.73994675192871591295422203349, 3.63451828696304477554439026319, 4.29596612740482499788865701474, 5.44760970385337951851209875175, 6.33612082012796205215323378261, 7.56226125394898655443539441061, 8.135099134881360610467066856944, 9.106576935860655901780955045682, 9.940716594533939444470739426263