L(s) = 1 | + (−3.70 − 7.08i)2-s + 15.5·3-s + (−36.5 + 52.5i)4-s + (108. + 61.7i)5-s + (−57.7 − 110. i)6-s − 629.·7-s + (507. + 64.1i)8-s + 243·9-s + (35.0 − 999. i)10-s − 1.25e3i·11-s + (−569. + 819. i)12-s − 3.33e3i·13-s + (2.33e3 + 4.45e3i)14-s + (1.69e3 + 962. i)15-s + (−1.42e3 − 3.83e3i)16-s − 7.63e3i·17-s + ⋯ |
L(s) = 1 | + (−0.463 − 0.886i)2-s + 0.577·3-s + (−0.570 + 0.821i)4-s + (0.869 + 0.494i)5-s + (−0.267 − 0.511i)6-s − 1.83·7-s + (0.992 + 0.125i)8-s + 0.333·9-s + (0.0350 − 0.999i)10-s − 0.943i·11-s + (−0.329 + 0.474i)12-s − 1.51i·13-s + (0.849 + 1.62i)14-s + (0.501 + 0.285i)15-s + (−0.348 − 0.937i)16-s − 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.219757 - 0.967518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.219757 - 0.967518i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.70 + 7.08i)T \) |
| 3 | \( 1 - 15.5T \) |
| 5 | \( 1 + (-108. - 61.7i)T \) |
good | 7 | \( 1 + 629.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.25e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.33e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.63e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.25e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.44e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 7.20e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.24e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.48e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 2.91e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.27e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.50e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.03e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.45e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.11e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.01e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.30e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.24e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 1.37e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.32e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.03e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.25e5iT - 8.32e11T^{2} \) |
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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
−13.34208018495080254323970207576, −12.39575058346857640017681342602, −10.69529466835355961111352050503, −9.836954523018978452963455563134, −9.095604920021937446338313118845, −7.49341607980856661203002635080, −5.89996678127426681033935096971, −3.37164098979780696342260359407, −2.67313579165604154683243461468, −0.43076385221055329937648443237,
1.83831352351837450226697748475, 4.18791127938713178838309307541, 6.08153762306009162230478616328, 6.88415225026001488314824277756, 8.602741429131448292546831667531, 9.611412216248188474007351400514, 10.06008749656047382355382473614, 12.57481314751059426457957959402, 13.38353834369989014373775078174, 14.36615004388143160621553150636