L(s) = 1 | − 2.44i·3-s + (1.73 + 1.41i)5-s + 1.41i·7-s − 2.99·9-s − 2·11-s + 5.65i·13-s + (3.46 − 4.24i)15-s − 4.89i·17-s + 6·19-s + 3.46·21-s − 7.07i·23-s + (0.999 + 4.89i)25-s + 6.92·29-s + 6.92·31-s + 4.89i·33-s + ⋯ |
L(s) = 1 | − 1.41i·3-s + (0.774 + 0.632i)5-s + 0.534i·7-s − 0.999·9-s − 0.603·11-s + 1.56i·13-s + (0.894 − 1.09i)15-s − 1.18i·17-s + 1.37·19-s + 0.755·21-s − 1.47i·23-s + (0.199 + 0.979i)25-s + 1.28·29-s + 1.24·31-s + 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.939237767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.939237767\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 4.24iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 2.44iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−9.550798164156332845800939297586, −8.702780181150424808332633057560, −7.80084632849986290489938123862, −6.83386254312843685526222133992, −6.63515393881984840420524307054, −5.61716972338139429731161729223, −4.62560460648512095216373261981, −2.77637623809755213216830777311, −2.38505760655077364117012964874, −1.10295173224343590109449848197,
1.08069263500393058636902178845, 2.84159582095365100080436358568, 3.72122957776554128220659963439, 4.75528904175595192242340437732, 5.39558164846142045528861163651, 6.03777946709549909170750957093, 7.54846083842040368174106890276, 8.257455103929111137938386713033, 9.173921882941455047692204809506, 9.970353241486313780353779952292