| L(s) = 1 | + i·3-s + (−1.67 + 1.48i)5-s + i·7-s − 9-s − 1.67·11-s + 0.869i·13-s + (−1.48 − 1.67i)15-s + 1.86i·17-s − 0.869·19-s − 21-s + i·23-s + (0.612 − 4.96i)25-s − i·27-s − 2.44·29-s − 4.19·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.749 + 0.662i)5-s + 0.377i·7-s − 0.333·9-s − 0.505·11-s + 0.241i·13-s + (−0.382 − 0.432i)15-s + 0.453i·17-s − 0.199·19-s − 0.218·21-s + 0.208i·23-s + (0.122 − 0.992i)25-s − 0.192i·27-s − 0.453·29-s − 0.753·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2448205778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2448205778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 0.869iT - 13T^{2} \) |
| 17 | \( 1 - 1.86iT - 17T^{2} \) |
| 19 | \( 1 + 0.869T + 19T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 + 2.76iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.11iT - 43T^{2} \) |
| 47 | \( 1 - 2.71iT - 47T^{2} \) |
| 53 | \( 1 + 3.63iT - 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 + 1.78T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 + 0.100T + 71T^{2} \) |
| 73 | \( 1 - 2.16iT - 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 + 9.09iT - 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 - 4.99iT - 97T^{2} \) |
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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
−10.24855552245154405640493815644, −9.231903607320071452735468939437, −8.499293571920891904724609941434, −7.68990592615887585851002756719, −6.88121085099408559555584303725, −5.90298994368192753163919035632, −5.01295195184554566015188705112, −3.97890346019327622627674007503, −3.25801566857281240583196657946, −2.11451482627319769464167510978,
0.099888995226492675145409627153, 1.41474282214450008622335414325, 2.83647520640854055661320221140, 3.90168852981352247993045995244, 4.87783537370614324363435415157, 5.67553448266041175115008612982, 6.85059159508438682670192574553, 7.50275312473501488240707229638, 8.214188757272070194807195782179, 8.880949712995550074492476356479
