L(s) = 1 | + 20.7i·2-s + (11.8 − 45.2i)3-s − 301.·4-s + 470.·5-s + (937. + 244. i)6-s − 481. i·7-s − 3.58e3i·8-s + (−1.90e3 − 1.06e3i)9-s + 9.75e3i·10-s − 6.88e3·11-s + (−3.55e3 + 1.36e4i)12-s − 1.06e4·13-s + 9.97e3·14-s + (5.56e3 − 2.13e4i)15-s + 3.57e4·16-s − 2.16e4·17-s + ⋯ |
L(s) = 1 | + 1.83i·2-s + (0.252 − 0.967i)3-s − 2.35·4-s + 1.68·5-s + (1.77 + 0.462i)6-s − 0.530i·7-s − 2.47i·8-s + (−0.872 − 0.489i)9-s + 3.08i·10-s − 1.55·11-s + (−0.594 + 2.27i)12-s − 1.34·13-s + 0.971·14-s + (0.425 − 1.63i)15-s + 2.18·16-s − 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0810 + 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0810 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.287283 - 0.264873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.287283 - 0.264873i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-11.8 + 45.2i)T \) |
| 23 | \( 1 + (5.50e4 - 1.92e4i)T \) |
good | 2 | \( 1 - 20.7iT - 128T^{2} \) |
| 5 | \( 1 - 470.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 481. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 6.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.06e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 7.53e3iT - 8.93e8T^{2} \) |
| 29 | \( 1 - 5.28e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.19e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.92e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 1.29e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 9.01e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.13e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 7.97e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.80e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.34e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 8.12e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 3.86e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 2.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.24e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 4.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.36e5iT - 8.07e13T^{2} \) |
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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
−13.45221042624184530390598254888, −12.77008787665100041031075205173, −10.23388963848591859108044385618, −9.178484436009620634796572644107, −7.909413496102348416154807463379, −7.01434543567751742351105149380, −5.96742695708611765422211178406, −5.04782180463846636910689879612, −2.25113288020595486421970967037, −0.11831170225732753508952113404,
2.30513965210136208617878839294, 2.58780140303189574672021037100, 4.65206365088228535155487345998, 5.52416967639428938237705508062, 8.556468941465811613541131329345, 9.643370146218275263100592342172, 10.09400566782568658218511751115, 10.97194919433230409045344659592, 12.40251144202351911547340718541, 13.40847674820695104520980874162