| L(s) = 1 | + 2·7-s − 3·9-s + 4·11-s − 2·13-s + 2·17-s − 19-s − 2·23-s − 5·25-s − 2·29-s + 8·31-s + 10·37-s + 6·41-s − 4·43-s + 2·47-s − 3·49-s − 2·53-s + 12·59-s + 8·61-s − 6·63-s + 4·67-s − 12·71-s + 10·73-s + 8·77-s − 4·79-s + 9·81-s + 4·83-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.229·19-s − 0.417·23-s − 25-s − 0.371·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s + 1.02·61-s − 0.755·63-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.911·77-s − 0.450·79-s + 81-s + 0.439·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.024065509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.024065509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.150379778452005476252038736491, −7.77971600808535577186472350431, −6.74948031493920360135961205789, −6.06450310550513585990973172718, −5.42006938312219904892482457282, −4.49664800353568894925274759606, −3.85563812432102534142768411561, −2.79681318110897758770813006745, −1.94306444844772002883091758432, −0.792034798490778959944484025778,
0.792034798490778959944484025778, 1.94306444844772002883091758432, 2.79681318110897758770813006745, 3.85563812432102534142768411561, 4.49664800353568894925274759606, 5.42006938312219904892482457282, 6.06450310550513585990973172718, 6.74948031493920360135961205789, 7.77971600808535577186472350431, 8.150379778452005476252038736491
