| L(s) = 1 | − 3-s − 5-s − 9-s + 11-s + 15-s − 5·17-s + 5·19-s − 23-s − 7·25-s + 4·29-s + 3·31-s − 33-s + 37-s − 4·41-s + 4·43-s + 45-s − 3·47-s − 7·49-s + 5·51-s − 7·53-s − 55-s − 5·57-s − 59-s − 61-s − 7·67-s + 69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1/3·9-s + 0.301·11-s + 0.258·15-s − 1.21·17-s + 1.14·19-s − 0.208·23-s − 7/5·25-s + 0.742·29-s + 0.538·31-s − 0.174·33-s + 0.164·37-s − 0.624·41-s + 0.609·43-s + 0.149·45-s − 0.437·47-s − 49-s + 0.700·51-s − 0.961·53-s − 0.134·55-s − 0.662·57-s − 0.130·59-s − 0.128·61-s − 0.855·67-s + 0.120·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 54 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 102 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.1554756221, −13.8288142064, −13.4674639139, −12.9525447933, −12.4014190892, −11.9525178904, −11.5827053976, −11.3394544027, −10.8987263309, −10.2629795507, −9.81043825035, −9.36093769531, −8.87334721007, −8.25096919094, −7.84671648552, −7.39932586608, −6.54486078015, −6.44781322045, −5.69537497044, −5.18755115662, −4.53816314606, −4.01550179748, −3.27986110507, −2.52512153348, −1.46765920048, 0,
1.46765920048, 2.52512153348, 3.27986110507, 4.01550179748, 4.53816314606, 5.18755115662, 5.69537497044, 6.44781322045, 6.54486078015, 7.39932586608, 7.84671648552, 8.25096919094, 8.87334721007, 9.36093769531, 9.81043825035, 10.2629795507, 10.8987263309, 11.3394544027, 11.5827053976, 11.9525178904, 12.4014190892, 12.9525447933, 13.4674639139, 13.8288142064, 14.1554756221
