| L(s) = 1 | + (0.744 + 1.20i)2-s + (2.39 − 0.209i)3-s + (−0.892 + 1.78i)4-s + (0.685 − 2.12i)5-s + (2.03 + 2.72i)6-s + (0.693 − 2.58i)7-s + (−2.81 + 0.258i)8-s + (2.72 − 0.479i)9-s + (3.06 − 0.759i)10-s + (3.81 + 2.20i)11-s + (−1.76 + 4.46i)12-s + (−5.36 − 0.469i)13-s + (3.62 − 1.09i)14-s + (1.19 − 5.23i)15-s + (−2.40 − 3.19i)16-s + (1.78 + 2.55i)17-s + ⋯ |
| L(s) = 1 | + (0.526 + 0.850i)2-s + (1.38 − 0.120i)3-s + (−0.446 + 0.894i)4-s + (0.306 − 0.951i)5-s + (0.829 + 1.11i)6-s + (0.262 − 0.978i)7-s + (−0.995 + 0.0912i)8-s + (0.907 − 0.159i)9-s + (0.970 − 0.240i)10-s + (1.14 + 0.663i)11-s + (−0.508 + 1.28i)12-s + (−1.48 − 0.130i)13-s + (0.970 − 0.291i)14-s + (0.308 − 1.35i)15-s + (−0.601 − 0.798i)16-s + (0.433 + 0.618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46757 + 0.842570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46757 + 0.842570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.744 - 1.20i)T \) |
| 5 | \( 1 + (-0.685 + 2.12i)T \) |
| 19 | \( 1 + (-0.355 - 4.34i)T \) |
| good | 3 | \( 1 + (-2.39 + 0.209i)T + (2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (-0.693 + 2.58i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.81 - 2.20i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.36 + 0.469i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.78 - 2.55i)T + (-5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (2.99 - 1.39i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (4.64 - 0.818i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.94 + 1.12i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.98 - 5.98i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.53 + 1.29i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.79 + 10.2i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.159 - 0.227i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.43 - 3.08i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.983 + 5.57i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.29 - 2.65i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (3.82 - 5.46i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (4.30 + 11.8i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.171 - 1.96i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-12.5 - 10.4i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.75 - 1.00i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.22 - 5.03i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.62 + 3.75i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−12.06255675792384507745931357562, −10.02637593710155956183359958180, −9.405781898304409968108993514970, −8.454800350565192231812796133195, −7.71998562519241123718322176970, −7.02161423294065853035739277108, −5.55614686015464506640680671605, −4.34452451532460621793245087640, −3.66361790470942495596243380747, −1.89446223936050182671494384615,
2.16294243586770793225022717267, 2.78791852976618103556130783676, 3.78942201888875311319475584480, 5.18754921008698967613596950437, 6.38981556503174942821898625881, 7.61319590715008026909634293858, 9.016999938808135895926645760240, 9.310478941789131727141961033418, 10.24475593423342554066421067029, 11.48468598244802317108782992558
