| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·13-s + 15-s − 6·17-s + 4·19-s + 21-s + 25-s − 27-s − 6·29-s + 4·31-s + 35-s + 2·37-s − 2·39-s + 6·41-s − 8·43-s − 45-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s + 12·59-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095985206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095985206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.314652961268928418960091821791, −8.647026888430366294854058821975, −7.64537707540215722485320944275, −6.91673575142178820615272805521, −6.16116790751690085616531757265, −5.30537393915954566746162190631, −4.32773350578479900565119001608, −3.54574519606089903353001249150, −2.27238370722428728570015712288, −0.74026101682650381544781632886,
0.74026101682650381544781632886, 2.27238370722428728570015712288, 3.54574519606089903353001249150, 4.32773350578479900565119001608, 5.30537393915954566746162190631, 6.16116790751690085616531757265, 6.91673575142178820615272805521, 7.64537707540215722485320944275, 8.647026888430366294854058821975, 9.314652961268928418960091821791
