| L(s) = 1 | + (−0.123 − 1.40i)2-s + (−0.0959 − 1.09i)3-s + (−1.96 + 0.348i)4-s + (−1.26 + 2.72i)5-s + (−1.53 + 0.270i)6-s + (2.14 + 3.72i)7-s + (0.734 + 2.73i)8-s + (1.76 − 0.310i)9-s + (3.99 + 1.45i)10-s + (−5.08 + 1.36i)11-s + (0.571 + 2.12i)12-s + (0.00310 − 0.0354i)13-s + (4.97 − 3.48i)14-s + (3.10 + 1.13i)15-s + (3.75 − 1.37i)16-s + (−1.16 + 6.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.0874 − 0.996i)2-s + (−0.0554 − 0.633i)3-s + (−0.984 + 0.174i)4-s + (−0.567 + 1.21i)5-s + (−0.626 + 0.110i)6-s + (0.812 + 1.40i)7-s + (0.259 + 0.965i)8-s + (0.586 − 0.103i)9-s + (1.26 + 0.459i)10-s + (−1.53 + 0.411i)11-s + (0.164 + 0.613i)12-s + (0.000861 − 0.00984i)13-s + (1.33 − 0.932i)14-s + (0.802 + 0.292i)15-s + (0.939 − 0.343i)16-s + (−0.283 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.910522 + 0.146818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.910522 + 0.146818i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.123 + 1.40i)T \) |
| 19 | \( 1 + (-0.221 + 4.35i)T \) |
| good | 3 | \( 1 + (0.0959 + 1.09i)T + (-2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (1.26 - 2.72i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-2.14 - 3.72i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.08 - 1.36i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.00310 + 0.0354i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (1.16 - 6.62i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (1.45 + 0.528i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-5.93 - 4.15i)T + (9.91 + 27.2i)T^{2} \) |
| 31 | \( 1 + (-1.11 - 1.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.47 + 4.47i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.48 + 1.25i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.89 - 0.884i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-7.02 + 1.23i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-5.50 - 11.8i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-4.54 - 6.48i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-1.29 - 2.78i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-4.84 + 6.92i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (2.84 + 7.83i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.32 - 2.77i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.50 + 7.97i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.51 + 0.405i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.626 - 0.525i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.30 - 1.64i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.87738984293193319148342070629, −10.70354875389439395427199008095, −10.45108136331925943511142836741, −8.856096883105562705807911806832, −8.051250721618304908956309181481, −7.14036260487605630652102691653, −5.68987356742610326218488769753, −4.45224117628035302645083336046, −2.86245035632565513510670167343, −2.01899493384890668439078513464,
0.73799039549596452261714666431, 3.94496358759881532841596233718, 4.73399445823050425871554498576, 5.27912073815104707698427741466, 7.11972133350773252662560165168, 7.896065958975002560501248043674, 8.478666822974185394133561682716, 9.859338271936324618361499300716, 10.37730449681624150866126022438, 11.62933364362420754551735822343
