| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s − 6·17-s + 6·19-s + 21-s + 8·23-s + 25-s − 27-s + 6·29-s + 6·31-s + 2·33-s + 35-s − 10·37-s + 2·41-s + 4·43-s − 45-s − 8·47-s + 49-s + 6·51-s − 12·53-s + 2·55-s − 6·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.348·33-s + 0.169·35-s − 1.64·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.64·53-s + 0.269·55-s − 0.794·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.220445638160044929496837312007, −7.37497321130478871197915841208, −6.76360727272757381454020390912, −6.09270049536733414349274513765, −4.92025339438886395369489047888, −4.72916901985484185209248139884, −3.39896233277134585610990366575, −2.71557781749145545205368216765, −1.24960858885676340498672394499, 0,
1.24960858885676340498672394499, 2.71557781749145545205368216765, 3.39896233277134585610990366575, 4.72916901985484185209248139884, 4.92025339438886395369489047888, 6.09270049536733414349274513765, 6.76360727272757381454020390912, 7.37497321130478871197915841208, 8.220445638160044929496837312007
