| L(s) = 1 | + i·3-s + (−0.599 + 2.15i)5-s − 0.0778i·7-s − 9-s − 4.03·11-s + 0.176i·13-s + (−2.15 − 0.599i)15-s + 7.72i·17-s − 1.88·19-s + 0.0778·21-s + 2.47i·23-s + (−4.28 − 2.58i)25-s − i·27-s + 6.84·29-s − 7.45·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.268 + 0.963i)5-s − 0.0294i·7-s − 0.333·9-s − 1.21·11-s + 0.0488i·13-s + (−0.556 − 0.154i)15-s + 1.87i·17-s − 0.432·19-s + 0.0169·21-s + 0.515i·23-s + (−0.856 − 0.516i)25-s − 0.192i·27-s + 1.27·29-s − 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1859373197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1859373197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.599 - 2.15i)T \) |
| 67 | \( 1 - iT \) |
| good | 7 | \( 1 + 0.0778iT - 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 - 0.176iT - 13T^{2} \) |
| 17 | \( 1 - 7.72iT - 17T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 23 | \( 1 - 2.47iT - 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 3.67iT - 37T^{2} \) |
| 41 | \( 1 + 0.0931T + 41T^{2} \) |
| 43 | \( 1 + 6.63iT - 43T^{2} \) |
| 47 | \( 1 + 8.43iT - 47T^{2} \) |
| 53 | \( 1 - 1.60iT - 53T^{2} \) |
| 59 | \( 1 + 0.795T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 0.963iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.08iT - 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.796992484009028702087313382832, −8.267499916001001351691692659505, −7.54541700204129545532988444004, −6.77212118069320989875177049413, −5.94833346776505449470519230839, −5.32867042811146742292087193282, −4.25640750808090075544942820042, −3.62861251164569507597899494318, −2.79525029800684158215477297810, −1.87238307920115725089148761476,
0.05838486879599218457472752514, 1.02069520164828616287942321101, 2.31105139330114217656546758676, 3.00529903325005597684213066868, 4.28131257198626721730702997246, 5.00661013722827384929203201184, 5.54138597114563527158638080685, 6.51126071677289583710164307996, 7.43250371310162499778012081650, 7.80892880040491032938278641609
