| L(s) = 1 | + 4·4-s − 5·5-s − 7·7-s + 13·11-s + 19·13-s + 16·16-s + 29·17-s − 20·20-s + 25·25-s − 28·28-s − 23·29-s + 35·35-s + 52·44-s − 31·47-s + 49·49-s + 76·52-s − 65·55-s + 64·64-s − 95·65-s + 116·68-s − 2·71-s + 34·73-s − 91·77-s − 157·79-s − 80·80-s + 86·83-s − 145·85-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s − 7-s + 1.18·11-s + 1.46·13-s + 16-s + 1.70·17-s − 20-s + 25-s − 28-s − 0.793·29-s + 35-s + 1.18·44-s − 0.659·47-s + 49-s + 1.46·52-s − 1.18·55-s + 64-s − 1.46·65-s + 1.70·68-s − 0.0281·71-s + 0.465·73-s − 1.18·77-s − 1.98·79-s − 80-s + 1.03·83-s − 1.70·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.813916491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.813916491\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 13 T + p^{2} T^{2} \) |
| 13 | \( 1 - 19 T + p^{2} T^{2} \) |
| 17 | \( 1 - 29 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 23 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 31 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( 1 + 2 T + p^{2} T^{2} \) |
| 73 | \( 1 - 34 T + p^{2} T^{2} \) |
| 79 | \( 1 + 157 T + p^{2} T^{2} \) |
| 83 | \( 1 - 86 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 149 T + p^{2} T^{2} \) |
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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
−11.53909938078069870630499054908, −10.69010714352588202670322378227, −9.636233618004979992010515391987, −8.500550253919827948579538127354, −7.48630701906346438116537150418, −6.60451494626098115725564219985, −5.78598926920184600444775710633, −3.81242820160127633922274281554, −3.24380662599066294345935222613, −1.19229059463459525297986245435,
1.19229059463459525297986245435, 3.24380662599066294345935222613, 3.81242820160127633922274281554, 5.78598926920184600444775710633, 6.60451494626098115725564219985, 7.48630701906346438116537150418, 8.500550253919827948579538127354, 9.636233618004979992010515391987, 10.69010714352588202670322378227, 11.53909938078069870630499054908
