| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 3·11-s − 14-s + 16-s + 20-s + 3·22-s + 4·23-s − 4·25-s + 28-s − 3·29-s + 5·31-s − 32-s + 35-s − 4·37-s − 40-s + 6·43-s − 3·44-s − 4·46-s − 12·47-s + 49-s + 4·50-s + 7·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s + 0.223·20-s + 0.639·22-s + 0.834·23-s − 4/5·25-s + 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.169·35-s − 0.657·37-s − 0.158·40-s + 0.914·43-s − 0.452·44-s − 0.589·46-s − 1.75·47-s + 1/7·49-s + 0.565·50-s + 0.961·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.09831387955213, −14.83601959075487, −14.12684182778217, −13.48333025351543, −13.17762055294939, −12.52637096200448, −11.84731538238784, −11.45774939209453, −10.73311909395726, −10.45136470769689, −9.822730601120129, −9.326678409124175, −8.765535357849614, −8.133250435083169, −7.729335044077277, −7.139041534456677, −6.480841690470668, −5.880410313975523, −5.251226163344540, −4.768399145845441, −3.857599455244869, −3.086241504956990, −2.410748021275531, −1.810354375356201, −0.9833536244631855, 0,
0.9833536244631855, 1.810354375356201, 2.410748021275531, 3.086241504956990, 3.857599455244869, 4.768399145845441, 5.251226163344540, 5.880410313975523, 6.480841690470668, 7.139041534456677, 7.729335044077277, 8.133250435083169, 8.765535357849614, 9.326678409124175, 9.822730601120129, 10.45136470769689, 10.73311909395726, 11.45774939209453, 11.84731538238784, 12.52637096200448, 13.17762055294939, 13.48333025351543, 14.12684182778217, 14.83601959075487, 15.09831387955213
