L(s) = 1 | + 0.298i·2-s − i·3-s + 1.91·4-s + (−1.68 − 1.47i)5-s + 0.298·6-s − 2.32i·7-s + 1.16i·8-s − 9-s + (0.439 − 0.501i)10-s − 1.91i·12-s − 4.59i·13-s + 0.691·14-s + (−1.47 + 1.68i)15-s + 3.47·16-s − 5.14i·17-s − 0.298i·18-s + ⋯ |
L(s) = 1 | + 0.210i·2-s − 0.577i·3-s + 0.955·4-s + (−0.751 − 0.659i)5-s + 0.121·6-s − 0.877i·7-s + 0.412i·8-s − 0.333·9-s + (0.138 − 0.158i)10-s − 0.551i·12-s − 1.27i·13-s + 0.184·14-s + (−0.380 + 0.434i)15-s + 0.868·16-s − 1.24i·17-s − 0.0702i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.570928153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570928153\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.68 + 1.47i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.298iT - 2T^{2} \) |
| 7 | \( 1 + 2.32iT - 7T^{2} \) |
| 13 | \( 1 + 4.59iT - 13T^{2} \) |
| 17 | \( 1 + 5.14iT - 17T^{2} \) |
| 19 | \( 1 - 5.69T + 19T^{2} \) |
| 23 | \( 1 - 6.02iT - 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 - 3.17iT - 37T^{2} \) |
| 41 | \( 1 + 0.468T + 41T^{2} \) |
| 43 | \( 1 + 8.08iT - 43T^{2} \) |
| 47 | \( 1 + 9.18iT - 47T^{2} \) |
| 53 | \( 1 + 4.25iT - 53T^{2} \) |
| 59 | \( 1 + 6.65T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 8.86iT - 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 9.08iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 8.34iT - 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 + 5.58iT - 97T^{2} \) |
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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
−8.809215811826255132092345484369, −7.75361195626458007182392663747, −7.43341261311097905734800282797, −7.03031554462724560740619456853, −5.52654800139796586425350447531, −5.30280072782043601484088011402, −3.71405532950931764325114885597, −3.08860150459106067456452959241, −1.62450361998969511796803246872, −0.55902054686764270028575643218,
1.76999676178350108995572768914, 2.75277747474838020453907756656, 3.58894332248835433772110975195, 4.40725084876517213589062029775, 5.70806281006975197585005789752, 6.31707053163034861419528883852, 7.21352150444939121530026778068, 7.88501669702713384279626133261, 8.890528495451628115192272406876, 9.548005112722549468861045632780