| L(s) = 1 | + (12.2 + 12.2i)7-s + 34.1i·11-s + (−25.8 + 25.8i)13-s + (−15.5 + 15.5i)17-s + 32.1i·19-s + (−72.9 − 72.9i)23-s − 106.·29-s + 10.3·31-s + (2.04 + 2.04i)37-s − 365. i·41-s + (10.2 − 10.2i)43-s + (−260. + 260. i)47-s − 40.4i·49-s + (42.4 + 42.4i)53-s + 375.·59-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.664i)7-s + 0.935i·11-s + (−0.550 + 0.550i)13-s + (−0.222 + 0.222i)17-s + 0.387i·19-s + (−0.661 − 0.661i)23-s − 0.681·29-s + 0.0601·31-s + (0.00907 + 0.00907i)37-s − 1.39i·41-s + (0.0363 − 0.0363i)43-s + (−0.806 + 0.806i)47-s − 0.118i·49-s + (0.109 + 0.109i)53-s + 0.828·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7355477314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7355477314\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-12.2 - 12.2i)T + 343iT^{2} \) |
| 11 | \( 1 - 34.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (25.8 - 25.8i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (15.5 - 15.5i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 32.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (72.9 + 72.9i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-2.04 - 2.04i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 365. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-10.2 + 10.2i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (260. - 260. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-42.4 - 42.4i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 770.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-603. - 603. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 695. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (262. - 262. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 32.2iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (519. + 519. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-416. - 416. i)T + 9.12e5iT^{2} \) |
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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
−10.05898478885260427467359709554, −9.285268668978776949694628386387, −8.457715351086873184642102676136, −7.61697271088366709437626039850, −6.75945117774552683785269693931, −5.71578369892106512529431284572, −4.81510371165033911531706716600, −3.98177471340021824064458423671, −2.44884047018337811216112154382, −1.68664288404074093097637775179,
0.18137728972185668504564017437, 1.42288740703780878532397523177, 2.80959505639460874708746041460, 3.88477256996369649619547178885, 4.90344430594120780244900057952, 5.75635070605353554853796528175, 6.83122994404244286190198072031, 7.74925438275826231377012628350, 8.315720503505692327713104177195, 9.394442625559234752155795689337
