| L(s) = 1 | − 2·2-s + 2·4-s − 10·13-s − 4·16-s − 6·17-s + 20·26-s + 8·32-s + 12·34-s − 10·37-s − 20·41-s − 20·52-s − 18·53-s − 24·61-s − 8·64-s − 12·68-s + 10·73-s + 20·74-s + 40·82-s + 10·97-s − 40·101-s + 36·106-s − 2·113-s + 22·121-s + 48·122-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.77·13-s − 16-s − 1.45·17-s + 3.92·26-s + 1.41·32-s + 2.05·34-s − 1.64·37-s − 3.12·41-s − 2.77·52-s − 2.47·53-s − 3.07·61-s − 64-s − 1.45·68-s + 1.17·73-s + 2.32·74-s + 4.41·82-s + 1.01·97-s − 3.98·101-s + 3.49·106-s − 0.188·113-s + 2·121-s + 4.34·122-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.809557666520366206216013992943, −9.435440952400528170353968722114, −9.186724986667811342583561262920, −8.740684199921409949362986918029, −8.150510840664915966133309281949, −7.992170970525122414646606191428, −7.24109136764166332693137310690, −7.22436923353086673792232802017, −6.55021710883288555029662046831, −6.45668935810405568703703915054, −5.38149730833813431252255983187, −4.85877680597085279192579707665, −4.81986722401855268615176217231, −4.12943534469073737174593173363, −3.17366072807781605341602982591, −2.75003568732305774571797246523, −1.86795042852037103120141142895, −1.75493628035455789074361619706, 0, 0,
1.75493628035455789074361619706, 1.86795042852037103120141142895, 2.75003568732305774571797246523, 3.17366072807781605341602982591, 4.12943534469073737174593173363, 4.81986722401855268615176217231, 4.85877680597085279192579707665, 5.38149730833813431252255983187, 6.45668935810405568703703915054, 6.55021710883288555029662046831, 7.22436923353086673792232802017, 7.24109136764166332693137310690, 7.992170970525122414646606191428, 8.150510840664915966133309281949, 8.740684199921409949362986918029, 9.186724986667811342583561262920, 9.435440952400528170353968722114, 9.809557666520366206216013992943
