| L(s) = 1 | − i·3-s + (2.21 + 0.315i)5-s − 0.0918i·7-s − 9-s − 3.06·11-s + 5.35i·13-s + (0.315 − 2.21i)15-s − 2.02i·17-s + 7.88·19-s − 0.0918·21-s − 5.50i·23-s + (4.80 + 1.39i)25-s + i·27-s − 2.34·29-s − 1.86·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.990 + 0.141i)5-s − 0.0347i·7-s − 0.333·9-s − 0.923·11-s + 1.48i·13-s + (0.0814 − 0.571i)15-s − 0.492i·17-s + 1.80·19-s − 0.0200·21-s − 1.14i·23-s + (0.960 + 0.279i)25-s + 0.192i·27-s − 0.435·29-s − 0.335·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245970564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245970564\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.21 - 0.315i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 + 0.0918iT - 7T^{2} \) |
| 11 | \( 1 + 3.06T + 11T^{2} \) |
| 13 | \( 1 - 5.35iT - 13T^{2} \) |
| 17 | \( 1 + 2.02iT - 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 + 5.50iT - 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 - 6.88iT - 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 + 6.19iT - 43T^{2} \) |
| 47 | \( 1 - 3.51iT - 47T^{2} \) |
| 53 | \( 1 + 1.50iT - 53T^{2} \) |
| 59 | \( 1 - 7.85T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 6.48iT - 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 - 3.43iT - 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.482642456440773516194242393028, −7.53912559928598752586365227139, −6.91063765796460228030642386795, −6.38569337090952876398296511585, −5.33846105950175498337387092027, −5.01676815549585183268366425966, −3.70685672979882029831063929323, −2.64433019345373162613132955043, −2.04770441183105429666464355002, −0.930621101673477384771255784771,
0.812959965645074319168758609190, 2.09670001557287879230279538323, 3.05054759036481549937507292321, 3.68378498917271482507744842828, 5.08226337726605185505685732733, 5.42326641902881677285774995897, 5.86588320455952925695819276656, 7.09954530035153848008023098724, 7.79537312366796050103623129187, 8.467201346295137872455597357499
