| L(s) = 1 | − 2-s + 4-s + 4·5-s + 7-s − 8-s − 4·10-s + 11-s − 6·13-s − 14-s + 16-s + 4·17-s − 2·19-s + 4·20-s − 22-s + 8·23-s + 11·25-s + 6·26-s + 28-s + 6·29-s + 6·31-s − 32-s − 4·34-s + 4·35-s − 6·37-s + 2·38-s − 4·40-s − 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s − 0.353·8-s − 1.26·10-s + 0.301·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.894·20-s − 0.213·22-s + 1.66·23-s + 11/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.676·35-s − 0.986·37-s + 0.324·38-s − 0.632·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780456472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780456472\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.739712159295331966490625255940, −8.917387992761630968547443589168, −8.140353505930131920153142154645, −6.96548548882838290586125297122, −6.54704897497816919862448175054, −5.33392541266418642466287088585, −4.91457124178222642670967255706, −3.01743557007021262394668003027, −2.19563344920726121984618617678, −1.16576401497843678501279592702,
1.16576401497843678501279592702, 2.19563344920726121984618617678, 3.01743557007021262394668003027, 4.91457124178222642670967255706, 5.33392541266418642466287088585, 6.54704897497816919862448175054, 6.96548548882838290586125297122, 8.140353505930131920153142154645, 8.917387992761630968547443589168, 9.739712159295331966490625255940
