| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s − 2·9-s + 2·10-s + 11-s − 12-s − 5·13-s + 2·15-s + 16-s + 2·18-s − 6·19-s − 2·20-s − 22-s + 24-s − 25-s + 5·26-s + 5·27-s + 6·29-s − 2·30-s + 4·31-s − 32-s − 33-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.38·13-s + 0.516·15-s + 1/4·16-s + 0.471·18-s − 1.37·19-s − 0.447·20-s − 0.213·22-s + 0.204·24-s − 1/5·25-s + 0.980·26-s + 0.962·27-s + 1.11·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s − 0.174·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.77760350964289, −15.31242488211246, −14.79134853578151, −14.33022552686127, −13.70102794253733, −12.81321924926839, −12.20653513541997, −12.01416736831272, −11.45340193733125, −10.96782768906498, −10.23656195933875, −10.03362301547526, −9.089293028303941, −8.609387412113704, −8.122947398478885, −7.577932601763565, −6.835000768089661, −6.489123048668935, −5.774094221007012, −4.946986320809222, −4.523252536436920, −3.682498941270960, −2.865324259900760, −2.280357332814378, −1.252928467099371, 0, 0,
1.252928467099371, 2.280357332814378, 2.865324259900760, 3.682498941270960, 4.523252536436920, 4.946986320809222, 5.774094221007012, 6.489123048668935, 6.835000768089661, 7.577932601763565, 8.122947398478885, 8.609387412113704, 9.089293028303941, 10.03362301547526, 10.23656195933875, 10.96782768906498, 11.45340193733125, 12.01416736831272, 12.20653513541997, 12.81321924926839, 13.70102794253733, 14.33022552686127, 14.79134853578151, 15.31242488211246, 15.77760350964289
