| L(s) = 1 | − 2·3-s − 5-s + 9-s − 3·11-s + 13-s + 2·15-s − 2·17-s + 19-s + 23-s + 25-s + 4·27-s + 2·29-s − 4·31-s + 6·33-s − 9·37-s − 2·39-s + 3·41-s + 2·43-s − 45-s + 9·47-s + 4·51-s − 9·53-s + 3·55-s − 2·57-s − 12·61-s − 65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.208·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s + 1.04·33-s − 1.47·37-s − 0.320·39-s + 0.468·41-s + 0.304·43-s − 0.149·45-s + 1.31·47-s + 0.560·51-s − 1.23·53-s + 0.404·55-s − 0.264·57-s − 1.53·61-s − 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.13939744336474, −15.79985182652765, −15.47073833958942, −14.63175549078883, −14.02242922508322, −13.41927990526005, −12.78949092246507, −12.19680272602826, −11.90762412537933, −11.03631501992742, −10.77273035590863, −10.41099269832867, −9.401880302387360, −8.898780438682435, −8.125910504204178, −7.566702064022970, −6.893235653648482, −6.277138594260872, −5.643578616515129, −5.056248993034244, −4.556930937229818, −3.640215476820005, −2.908888505306928, −1.957972137328605, −0.8252945448632490, 0,
0.8252945448632490, 1.957972137328605, 2.908888505306928, 3.640215476820005, 4.556930937229818, 5.056248993034244, 5.643578616515129, 6.277138594260872, 6.893235653648482, 7.566702064022970, 8.125910504204178, 8.898780438682435, 9.401880302387360, 10.41099269832867, 10.77273035590863, 11.03631501992742, 11.90762412537933, 12.19680272602826, 12.78949092246507, 13.41927990526005, 14.02242922508322, 14.63175549078883, 15.47073833958942, 15.79985182652765, 16.13939744336474
