| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 2·13-s + 15-s + 16-s + 4·17-s − 18-s + 6·19-s − 20-s + 2·22-s + 6·23-s + 24-s + 25-s − 2·26-s − 27-s − 10·29-s − 30-s + 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.426·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 1.85·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494781229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494781229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.31564364287787, −13.65000617887993, −13.16571274848709, −12.69915224964031, −12.04790811502051, −11.49924006339115, −11.38851099042690, −10.66355382874786, −10.16221987137455, −9.781121578201119, −9.108337403429809, −8.636459937043272, −7.955144875802833, −7.492554972020785, −7.162027184137464, −6.486882002357964, −5.685794080480322, −5.456849986337633, −4.776312638183112, −3.964409085940149, −3.290096723638002, −2.837280646791490, −1.856282586635236, −1.031973056208246, −0.5972919228674764,
0.5972919228674764, 1.031973056208246, 1.856282586635236, 2.837280646791490, 3.290096723638002, 3.964409085940149, 4.776312638183112, 5.456849986337633, 5.685794080480322, 6.486882002357964, 7.162027184137464, 7.492554972020785, 7.955144875802833, 8.636459937043272, 9.108337403429809, 9.781121578201119, 10.16221987137455, 10.66355382874786, 11.38851099042690, 11.49924006339115, 12.04790811502051, 12.69915224964031, 13.16571274848709, 13.65000617887993, 14.31564364287787
