| L(s) = 1 | + (−0.723 + 1.21i)2-s + (0.987 + 0.156i)3-s + (−0.951 − 1.75i)4-s + (−1.94 + 1.09i)5-s + (−0.905 + 1.08i)6-s + (−2.52 + 2.52i)7-s + (2.82 + 0.117i)8-s + (0.951 + 0.309i)9-s + (0.0755 − 3.16i)10-s + (−1.68 + 0.548i)11-s + (−0.664 − 1.88i)12-s + (−1.62 − 3.19i)13-s + (−1.23 − 4.89i)14-s + (−2.09 + 0.780i)15-s + (−2.18 + 3.34i)16-s + (0.815 + 5.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.511 + 0.859i)2-s + (0.570 + 0.0903i)3-s + (−0.475 − 0.879i)4-s + (−0.871 + 0.491i)5-s + (−0.369 + 0.443i)6-s + (−0.954 + 0.954i)7-s + (0.999 + 0.0415i)8-s + (0.317 + 0.103i)9-s + (0.0238 − 0.999i)10-s + (−0.509 + 0.165i)11-s + (−0.191 − 0.544i)12-s + (−0.451 − 0.885i)13-s + (−0.331 − 1.30i)14-s + (−0.541 + 0.201i)15-s + (−0.547 + 0.837i)16-s + (0.197 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0413423 - 0.479118i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0413423 - 0.479118i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.723 - 1.21i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| good | 7 | \( 1 + (2.52 - 2.52i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.68 - 0.548i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.62 + 3.19i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.815 - 5.14i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (6.47 + 4.70i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.63 - 7.12i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-2.24 - 3.09i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.52 - 3.47i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 2.07i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.90 + 5.86i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.90 - 6.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.113 - 0.716i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (0.150 - 0.949i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (0.117 - 0.360i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.14 - 9.67i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 + 2.14i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-0.772 - 1.06i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.731 - 0.372i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (5.19 - 3.77i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.109 - 0.694i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (4.92 - 1.60i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.43 - 1.33i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.52576108490102934110795002439, −10.95659445776221177571876290499, −10.19529399339776081430595837095, −9.193858577914267169349917853481, −8.342296379328457470277224236818, −7.57557587689778293741238656556, −6.54413993359065802344475711376, −5.50333318489676670411643259923, −4.00582070646050641770723493009, −2.60073383640850936732437564478,
0.37502584893758065118913584891, 2.46141941630646909185272954427, 3.82557407787816939713893526801, 4.46937739764139869665836330828, 6.71247048885142598828090981647, 7.71083761233192126069447042842, 8.428402801685866649637200563626, 9.499479550746577398058577098404, 10.19417640324061184137977195910, 11.19938046356051882933434048878
