L(s) = 1 | + i·3-s − 2.23·5-s + i·7-s − 9-s − 2·11-s − 4.47i·13-s − 2.23i·15-s − 2.47i·17-s − 2·19-s − 21-s − 4i·23-s + 5.00·25-s − i·27-s − 0.472·29-s + 8.47·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.999·5-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s − 1.24i·13-s − 0.577i·15-s − 0.599i·17-s − 0.458·19-s − 0.218·21-s − 0.834i·23-s + 1.00·25-s − 0.192i·27-s − 0.0876·29-s + 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.446376 - 0.446376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.446376 - 0.446376i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 6.47iT - 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47iT - 43T^{2} \) |
| 47 | \( 1 + 2.47iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 16.4iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−10.28682495112734319309462724443, −9.012718258494766195010397558756, −8.310296079345296557464064391804, −7.63985954109313844049830889064, −6.52103682168705204079475098304, −5.34754315983756251486129360662, −4.64502229655117892086657898728, −3.49835591600229882481778576195, −2.61803696141545701228845624429, −0.31542836009170264873492228904,
1.47624729189004901500114460180, 2.96406642836973722026363561312, 4.08375880049717079489989322663, 4.94458397330360784046541134541, 6.35635006666296177188568674581, 6.97111860677075305927503677494, 7.959483524688464109005638000150, 8.406599207245090154433218478009, 9.548105050050946626604603968784, 10.54138937776846917398213475471