| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 6·11-s + 2·13-s − 15-s − 2·17-s − 2·19-s − 2·21-s − 6·23-s + 25-s − 27-s − 10·29-s + 4·31-s + 6·33-s + 2·35-s − 2·37-s − 2·39-s + 6·41-s + 45-s − 6·47-s − 3·49-s + 2·51-s + 10·53-s − 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 1.04·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s − 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.712784544466545781669039199835, −7.978821379245276786308433262806, −7.35263666430954002912709522736, −6.20594403409908280738398626986, −5.60208559701857339203852080548, −4.88862007191587476774362230153, −3.98117379548078477179497732925, −2.57894927094631232027358788260, −1.68720638789308651755146426845, 0,
1.68720638789308651755146426845, 2.57894927094631232027358788260, 3.98117379548078477179497732925, 4.88862007191587476774362230153, 5.60208559701857339203852080548, 6.20594403409908280738398626986, 7.35263666430954002912709522736, 7.978821379245276786308433262806, 8.712784544466545781669039199835
