| L(s) = 1 | − 5-s − 7-s + 6·11-s + 6·13-s − 2·17-s − 7·19-s + 23-s − 4·25-s + 2·29-s − 10·31-s + 35-s − 6·37-s − 8·41-s + 10·43-s − 8·47-s + 49-s + 2·53-s − 6·55-s + 7·61-s − 6·65-s + 12·67-s − 15·71-s − 2·73-s − 6·77-s − 79-s − 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.80·11-s + 1.66·13-s − 0.485·17-s − 1.60·19-s + 0.208·23-s − 4/5·25-s + 0.371·29-s − 1.79·31-s + 0.169·35-s − 0.986·37-s − 1.24·41-s + 1.52·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.809·55-s + 0.896·61-s − 0.744·65-s + 1.46·67-s − 1.78·71-s − 0.234·73-s − 0.683·77-s − 0.112·79-s − 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.22465275451459589553360512613, −6.60256748385297177905641829045, −6.24011840284485727614740106009, −5.44460156464098153746842202222, −4.21537144300461810412728058001, −3.93503192386312453991018439618, −3.33970214873432747677355733449, −2.01373441152095988198763482501, −1.30267713673448821438754961046, 0,
1.30267713673448821438754961046, 2.01373441152095988198763482501, 3.33970214873432747677355733449, 3.93503192386312453991018439618, 4.21537144300461810412728058001, 5.44460156464098153746842202222, 6.24011840284485727614740106009, 6.60256748385297177905641829045, 7.22465275451459589553360512613
