| L(s) = 1 | + (9.68 − 9.68i)3-s + (48.6 + 48.6i)7-s + 55.4i·9-s − 463. i·11-s + (−320. − 320. i)13-s + (−1.04e3 + 1.04e3i)17-s − 701.·19-s + 942.·21-s + (−2.00e3 + 2.00e3i)23-s + (2.89e3 + 2.89e3i)27-s + 3.56e3i·29-s − 9.04e3i·31-s + (−4.48e3 − 4.48e3i)33-s + (−1.64e3 + 1.64e3i)37-s − 6.21e3·39-s + ⋯ |
| L(s) = 1 | + (0.621 − 0.621i)3-s + (0.375 + 0.375i)7-s + 0.228i·9-s − 1.15i·11-s + (−0.526 − 0.526i)13-s + (−0.877 + 0.877i)17-s − 0.445·19-s + 0.466·21-s + (−0.789 + 0.789i)23-s + (0.762 + 0.762i)27-s + 0.787i·29-s − 1.69i·31-s + (−0.716 − 0.716i)33-s + (−0.197 + 0.197i)37-s − 0.654·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.02100615145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02100615145\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-9.68 + 9.68i)T - 243iT^{2} \) |
| 7 | \( 1 + (-48.6 - 48.6i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 463. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (320. + 320. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.04e3 - 1.04e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 701.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.00e3 - 2.00e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 3.56e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.04e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.64e3 - 1.64e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-3.94e3 + 3.94e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (7.94e3 + 7.94e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.16e4 + 1.16e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.12e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-9.19e3 - 9.19e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.26e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.79e4 - 2.79e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 8.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (7.72e4 - 7.72e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.45e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.78e4 - 9.78e4i)T - 8.58e9iT^{2} \) |
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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
−10.92964238068960496653968349447, −9.919932341481499773749474548912, −8.573489641115087663539364870563, −8.305753630562646721448249021906, −7.27749395853454174432171633015, −6.12779576255625900512413359201, −5.15785183212453201661501315840, −3.74013842776824364616878840895, −2.52845074766209920346544441505, −1.60971863199526786138454127384,
0.00422404471715352567082216762, 1.78497371246612500620981810898, 2.92522965167848775750250944894, 4.34182295672174015209675719432, 4.70425093396852898314450106655, 6.45014532815624576498018125743, 7.26917158245342170483740273441, 8.382172658409905132469235886377, 9.267533228291862309765815753721, 9.933107117940101871811783891453
