| L(s) = 1 | + (0.401 + 1.35i)2-s + (0.242 − 2.77i)3-s + (−1.67 + 1.08i)4-s + (−0.947 − 2.03i)5-s + (3.85 − 0.784i)6-s + (−0.0320 + 0.0554i)7-s + (−2.14 − 1.83i)8-s + (−4.68 − 0.825i)9-s + (2.37 − 2.10i)10-s + (−4.04 − 1.08i)11-s + (2.61 + 4.91i)12-s + (−0.161 − 1.84i)13-s + (−0.0880 − 0.0211i)14-s + (−5.86 + 2.13i)15-s + (1.62 − 3.65i)16-s + (1.04 + 5.91i)17-s + ⋯ |
| L(s) = 1 | + (0.283 + 0.958i)2-s + (0.140 − 1.60i)3-s + (−0.838 + 0.544i)4-s + (−0.423 − 0.908i)5-s + (1.57 − 0.320i)6-s + (−0.0121 + 0.0209i)7-s + (−0.760 − 0.649i)8-s + (−1.56 − 0.275i)9-s + (0.751 − 0.664i)10-s + (−1.22 − 0.327i)11-s + (0.754 + 1.41i)12-s + (−0.0448 − 0.512i)13-s + (−0.0235 − 0.00565i)14-s + (−1.51 + 0.551i)15-s + (0.407 − 0.913i)16-s + (0.252 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.688977 - 0.751319i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688977 - 0.751319i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.401 - 1.35i)T \) |
| 19 | \( 1 + (1.50 + 4.08i)T \) |
| good | 3 | \( 1 + (-0.242 + 2.77i)T + (-2.95 - 0.520i)T^{2} \) |
| 5 | \( 1 + (0.947 + 2.03i)T + (-3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (0.0320 - 0.0554i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.04 + 1.08i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.161 + 1.84i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 5.91i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-7.10 + 2.58i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.17 + 1.52i)T + (9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (-2.86 + 4.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.41 - 2.41i)T - 37iT^{2} \) |
| 41 | \( 1 + (4.08 - 3.42i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.35 + 3.89i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.34 + 0.237i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.0696 + 0.149i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (7.60 - 10.8i)T + (-20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (-3.44 + 7.38i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (0.176 + 0.252i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-3.29 + 9.04i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.130 + 0.155i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.79 + 4.86i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (14.1 - 3.78i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.47 - 1.23i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (5.12 - 0.903i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.04587399656041676966772366737, −10.64297204020228816415222332964, −8.924426457790811201062045972179, −8.223716505498914525067774044407, −7.72473090002427876825606644610, −6.65401745297425890891925878078, −5.70841900190306967466170259093, −4.61171137861531695361752279917, −2.83040079722737669224582024098, −0.67990630625933793195019187583,
2.75422356933032235814551018027, 3.47944725730719880174237433185, 4.68331125317182398936536029167, 5.38853028231232509625698442948, 7.18113172544717017326758647929, 8.611427447326198025079529611937, 9.561839662627848326176748517146, 10.28549492331136243152920650670, 10.88447906069400089915174403533, 11.57724252850997201883752722624
