| L(s) = 1 | − 3-s − 0.618·7-s − 2·9-s + 5.23·11-s + 1.85·13-s − 5.23·17-s − 0.854·19-s + 0.618·21-s − 3.76·23-s + 5·27-s − 3.61·29-s + 3·31-s − 5.23·33-s − 0.236·37-s − 1.85·39-s − 0.763·41-s + 4.85·43-s − 0.618·47-s − 6.61·49-s + 5.23·51-s − 3.47·53-s + 0.854·57-s + 10.8·59-s + 8.70·61-s + 1.23·63-s − 4.76·67-s + 3.76·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.233·7-s − 0.666·9-s + 1.57·11-s + 0.514·13-s − 1.26·17-s − 0.195·19-s + 0.134·21-s − 0.784·23-s + 0.962·27-s − 0.671·29-s + 0.538·31-s − 0.911·33-s − 0.0388·37-s − 0.296·39-s − 0.119·41-s + 0.740·43-s − 0.0901·47-s − 0.945·49-s + 0.733·51-s − 0.476·53-s + 0.113·57-s + 1.41·59-s + 1.11·61-s + 0.155·63-s − 0.582·67-s + 0.453·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287210241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287210241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 0.236T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
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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
−7.58256157592927554216971391934, −6.61970694315756961986379481611, −6.41883430976531365165324233420, −5.78207333507983897116187494921, −4.92056851583725871929724767066, −4.11897141050590023814370146048, −3.60134250862243611148278528056, −2.54308627954451348977099643290, −1.64079454363098128415151813266, −0.55786222492347363933738490197,
0.55786222492347363933738490197, 1.64079454363098128415151813266, 2.54308627954451348977099643290, 3.60134250862243611148278528056, 4.11897141050590023814370146048, 4.92056851583725871929724767066, 5.78207333507983897116187494921, 6.41883430976531365165324233420, 6.61970694315756961986379481611, 7.58256157592927554216971391934
