| L(s) = 1 | − i·3-s + (−2.22 + 0.196i)5-s + 0.128i·7-s − 9-s + 0.0389·11-s − 3.81i·13-s + (0.196 + 2.22i)15-s + 6.63i·17-s + 7.30·19-s + 0.128·21-s − 3.64i·23-s + (4.92 − 0.874i)25-s + i·27-s − 8.25·29-s − 7.07·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.996 + 0.0877i)5-s + 0.0484i·7-s − 0.333·9-s + 0.0117·11-s − 1.05i·13-s + (0.0506 + 0.575i)15-s + 1.61i·17-s + 1.67·19-s + 0.0279·21-s − 0.760i·23-s + (0.984 − 0.174i)25-s + 0.192i·27-s − 1.53·29-s − 1.27·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4818936313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4818936313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.22 - 0.196i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.128iT - 7T^{2} \) |
| 11 | \( 1 - 0.0389T + 11T^{2} \) |
| 13 | \( 1 + 3.81iT - 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 7.30T + 19T^{2} \) |
| 23 | \( 1 + 3.64iT - 23T^{2} \) |
| 29 | \( 1 + 8.25T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 7.37iT - 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 7.72iT - 43T^{2} \) |
| 47 | \( 1 - 9.62iT - 47T^{2} \) |
| 53 | \( 1 + 3.82iT - 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 + 0.309iT - 73T^{2} \) |
| 79 | \( 1 + 5.72T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.10iT - 97T^{2} \) |
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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
−7.79349100138725302024660829997, −7.61188734425633525235756564573, −6.79456727860522899945176441028, −5.74210188381963398742446657693, −5.34272797929525699975086512922, −3.99986485153594544459537675384, −3.53728295655674798426571014404, −2.52302740711017787918785386685, −1.32183677331753648790825581456, −0.15415128468608667053240376003,
1.26047582164221069277939420141, 2.70698768380421271903567466828, 3.54425498816033074318323736309, 4.15213626918606380001585070050, 5.09053003920086898807316188886, 5.52975331715437766184696760643, 6.91217158868810171671435778468, 7.29809592936895279048971720150, 7.980967179520593041333295113398, 9.028986782554265495047851790792
