| L(s) = 1 | − 5.83·5-s − 5.24·7-s + 1.24·11-s − 5.89i·13-s + 1.57·17-s + (−10.6 − 15.7i)19-s + 27.5·23-s + 9.05·25-s − 15.9i·29-s − 53.2i·31-s + 30.5·35-s − 10.0i·37-s + 69.8i·41-s + 52.9·43-s + 12.2·47-s + ⋯ |
| L(s) = 1 | − 1.16·5-s − 0.748·7-s + 0.112·11-s − 0.453i·13-s + 0.0923·17-s + (−0.561 − 0.827i)19-s + 1.19·23-s + 0.362·25-s − 0.548i·29-s − 1.71i·31-s + 0.873·35-s − 0.270i·37-s + 1.70i·41-s + 1.23·43-s + 0.259·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2228908460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2228908460\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (10.6 + 15.7i)T \) |
| good | 5 | \( 1 + 5.83T + 25T^{2} \) |
| 7 | \( 1 + 5.24T + 49T^{2} \) |
| 11 | \( 1 - 1.24T + 121T^{2} \) |
| 13 | \( 1 + 5.89iT - 169T^{2} \) |
| 17 | \( 1 - 1.57T + 289T^{2} \) |
| 23 | \( 1 - 27.5T + 529T^{2} \) |
| 29 | \( 1 + 15.9iT - 841T^{2} \) |
| 31 | \( 1 + 53.2iT - 961T^{2} \) |
| 37 | \( 1 + 10.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 69.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 12.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 40.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 75.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 47.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 56.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 24.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.19iT - 9.40e3T^{2} \) |
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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
−8.979369197738097469232974491848, −7.939404869470603155026245107094, −7.58523155159474271494075794703, −6.62952390869300265873883597556, −5.98971375928049535054418851045, −4.83739310105648149402106391520, −4.13375362361393340208140290237, −3.29520830092816607162755325730, −2.49732912416342910575139579279, −0.830166879781777634386541149546,
0.07135233406288597250521606954, 1.38332283145928466397597991945, 2.80678129845427368167642892147, 3.61126994928244494636122501296, 4.25007238098037271984229123670, 5.22662612328914117147902235119, 6.18555966214596178013893574257, 7.06812429374037131161040913604, 7.42937470803155453352620421356, 8.608620312754440426382853164750
