| L(s) = 1 | + i·3-s + (0.853 − 2.06i)5-s − 2.56i·7-s − 9-s + 0.836·11-s − 5.01i·13-s + (2.06 + 0.853i)15-s + 1.93i·17-s − 1.56·19-s + 2.56·21-s + 2.41i·23-s + (−3.54 − 3.52i)25-s − i·27-s + 6.17·29-s + 4.81·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.381 − 0.924i)5-s − 0.969i·7-s − 0.333·9-s + 0.252·11-s − 1.39i·13-s + (0.533 + 0.220i)15-s + 0.469i·17-s − 0.358·19-s + 0.559·21-s + 0.504i·23-s + (−0.708 − 0.705i)25-s − 0.192i·27-s + 1.14·29-s + 0.864·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584744859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584744859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.853 + 2.06i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 + 2.56iT - 7T^{2} \) |
| 11 | \( 1 - 0.836T + 11T^{2} \) |
| 13 | \( 1 + 5.01iT - 13T^{2} \) |
| 17 | \( 1 - 1.93iT - 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 2.41iT - 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 - 2.01iT - 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 6.76iT - 47T^{2} \) |
| 53 | \( 1 - 6.96iT - 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 2.05iT - 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 + 2.67iT - 97T^{2} \) |
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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
−8.420826825343420128870981725739, −7.61460234053411137672010444281, −6.70824900703305598431641892805, −5.81320600197697567118611148486, −5.20027710019767111484191288392, −4.39685939191142899752967595444, −3.76077357875125998906752105364, −2.77327308371611279739971790965, −1.43953589070796916739520898475, −0.45908627293434627518347142876,
1.41928393092254485104038057950, 2.40576069814929837136622879211, 2.86684017312892015486012335909, 4.10268994455610225957834144709, 5.00985430011357867753187235969, 5.98819356660220008344098250940, 6.56382579964168865294616608374, 6.92827676657983152848055676121, 7.967158003967478630921610763505, 8.651055664702824426087988583947
