| L(s) = 1 | + (0.742 − 0.117i)3-s + (4.02 − 2.96i)5-s + (−2.71 − 2.71i)7-s + (−8.02 + 2.60i)9-s + (4.47 − 13.7i)11-s + (−16.6 − 8.47i)13-s + (2.64 − 2.67i)15-s + (10.5 + 1.67i)17-s + (−11.3 − 15.6i)19-s + (−2.33 − 1.69i)21-s + (8.15 + 16.0i)23-s + (7.45 − 23.8i)25-s + (−11.6 + 5.95i)27-s + (−15.1 + 20.8i)29-s + (−6.46 + 4.70i)31-s + ⋯ |
| L(s) = 1 | + (0.247 − 0.0392i)3-s + (0.805 − 0.592i)5-s + (−0.387 − 0.387i)7-s + (−0.891 + 0.289i)9-s + (0.407 − 1.25i)11-s + (−1.27 − 0.651i)13-s + (0.176 − 0.178i)15-s + (0.621 + 0.0985i)17-s + (−0.597 − 0.822i)19-s + (−0.111 − 0.0807i)21-s + (0.354 + 0.695i)23-s + (0.298 − 0.954i)25-s + (−0.432 + 0.220i)27-s + (−0.523 + 0.720i)29-s + (−0.208 + 0.151i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.822402 - 1.19571i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822402 - 1.19571i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.02 + 2.96i)T \) |
| good | 3 | \( 1 + (-0.742 + 0.117i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (2.71 + 2.71i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.47 + 13.7i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (16.6 + 8.47i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-10.5 - 1.67i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (11.3 + 15.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-8.15 - 16.0i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (15.1 - 20.8i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (6.46 - 4.70i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-31.9 + 62.7i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 3.72i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (9.83 - 9.83i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.34 + 33.7i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (3.64 - 0.577i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-111. + 36.1i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-9.28 + 28.5i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-62.2 - 9.85i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (94.8 + 68.9i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-35.3 - 69.4i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (85.0 - 117. i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-1.65 + 10.4i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-3.88 - 1.26i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 105. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74974763547374110932076677713, −9.743583113717470578914359405270, −8.984384032866919113143911274598, −8.169958947231781925235545167654, −7.01239952683318234513093443172, −5.77256136840539954266005641937, −5.16684674347979325252026168871, −3.52821800832981082480726241761, −2.38692523795045589858908996067, −0.57810632588036344533231831884,
2.03383790018550535169904569103, 2.93412924688740409421052055324, 4.45068732901159544891711562638, 5.73125172167895311980963503465, 6.57519630364359621303140381027, 7.50457479456862450483122797562, 8.784866466463713084579193032975, 9.741736326060411539798582068154, 10.02721641257951067082047902001, 11.45233042437363794650327172872
