| L(s) = 1 | − 1.20·2-s − 3-s − 0.536·4-s + 5-s + 1.20·6-s − 3.03·7-s + 3.06·8-s + 9-s − 1.20·10-s + 5.64·11-s + 0.536·12-s + 0.245·13-s + 3.67·14-s − 15-s − 2.63·16-s + 6.39·17-s − 1.20·18-s − 7.32·19-s − 0.536·20-s + 3.03·21-s − 6.82·22-s + 7.15·23-s − 3.06·24-s + 25-s − 0.296·26-s − 27-s + 1.62·28-s + ⋯ |
| L(s) = 1 | − 0.855·2-s − 0.577·3-s − 0.268·4-s + 0.447·5-s + 0.493·6-s − 1.14·7-s + 1.08·8-s + 0.333·9-s − 0.382·10-s + 1.70·11-s + 0.154·12-s + 0.0679·13-s + 0.981·14-s − 0.258·15-s − 0.659·16-s + 1.55·17-s − 0.285·18-s − 1.68·19-s − 0.119·20-s + 0.662·21-s − 1.45·22-s + 1.49·23-s − 0.626·24-s + 0.200·25-s − 0.0581·26-s − 0.192·27-s + 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8312574104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8312574104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 163 | \( 1 + T \) |
| good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 0.245T + 13T^{2} \) |
| 17 | \( 1 - 6.39T + 17T^{2} \) |
| 19 | \( 1 + 7.32T + 19T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 + 0.0229T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 2.27T + 67T^{2} \) |
| 71 | \( 1 + 0.597T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 0.855T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.991907166072088356680196905624, −8.529370075918102871448999889542, −7.28045582819686262135834100084, −6.67476630920956265072440815562, −6.06427436923593897666109886229, −5.05683300901566188296032838753, −4.10086204817678285284347494630, −3.26227249400073630955771231142, −1.66420392359287521238324455893, −0.72744153075780961847668942150,
0.72744153075780961847668942150, 1.66420392359287521238324455893, 3.26227249400073630955771231142, 4.10086204817678285284347494630, 5.05683300901566188296032838753, 6.06427436923593897666109886229, 6.67476630920956265072440815562, 7.28045582819686262135834100084, 8.529370075918102871448999889542, 8.991907166072088356680196905624
