| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 4·11-s − 6·13-s − 15-s − 2·17-s + 4·19-s − 21-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 35-s − 6·37-s + 6·39-s + 6·41-s + 8·43-s + 45-s + 4·47-s + 49-s + 2·51-s + 2·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.986·37-s + 0.960·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.50484224554463017477857097769, −7.05595107945793261432040671799, −6.10945243198473086516229657460, −5.43050421089799724978737845000, −4.85435021154571407162080225304, −4.33410151894050894090402657081, −2.82370190508741805825161401190, −2.47148123535568267642924242454, −1.20946982934137193450784731388, 0,
1.20946982934137193450784731388, 2.47148123535568267642924242454, 2.82370190508741805825161401190, 4.33410151894050894090402657081, 4.85435021154571407162080225304, 5.43050421089799724978737845000, 6.10945243198473086516229657460, 7.05595107945793261432040671799, 7.50484224554463017477857097769
