| L(s) = 1 | − 3-s − 4-s + 9-s − 11-s + 12-s + 16-s + 4·25-s − 27-s + 33-s − 36-s − 5·37-s − 41-s + 44-s − 48-s − 13·49-s + 8·53-s − 64-s + 10·67-s + 71-s − 16·73-s − 4·75-s + 81-s − 19·83-s − 99-s − 4·100-s + 9·101-s − 15·107-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1/4·16-s + 4/5·25-s − 0.192·27-s + 0.174·33-s − 1/6·36-s − 0.821·37-s − 0.156·41-s + 0.150·44-s − 0.144·48-s − 1.85·49-s + 1.09·53-s − 1/8·64-s + 1.22·67-s + 0.118·71-s − 1.87·73-s − 0.461·75-s + 1/9·81-s − 2.08·83-s − 0.100·99-s − 2/5·100-s + 0.895·101-s − 1.45·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 37 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 123 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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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
−8.479949642814950153298403937031, −7.939179028743262748044438962742, −7.42219394335140015680744675198, −6.87437117393331103640111423399, −6.61858976304572735198780795604, −5.95286802142354630135508139838, −5.39179250578626559447901397982, −5.19081358036997044474537185209, −4.47190375665545940502180349363, −4.15657981246669833852968742857, −3.35704156534283470621964569579, −2.88129273721453435437442319229, −1.96059803051084060688896860349, −1.14010213152465505162363804303, 0,
1.14010213152465505162363804303, 1.96059803051084060688896860349, 2.88129273721453435437442319229, 3.35704156534283470621964569579, 4.15657981246669833852968742857, 4.47190375665545940502180349363, 5.19081358036997044474537185209, 5.39179250578626559447901397982, 5.95286802142354630135508139838, 6.61858976304572735198780795604, 6.87437117393331103640111423399, 7.42219394335140015680744675198, 7.939179028743262748044438962742, 8.479949642814950153298403937031
