| L(s) = 1 | − 4-s − 4·7-s + 2·13-s − 3·16-s − 4·19-s − 7·25-s + 4·28-s + 16·31-s − 14·37-s − 4·43-s − 2·49-s − 2·52-s + 14·61-s + 7·64-s − 20·67-s + 14·73-s + 4·76-s − 4·79-s − 8·91-s + 4·97-s + 7·100-s + 16·103-s − 22·109-s + 12·112-s − 16·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s + 0.554·13-s − 3/4·16-s − 0.917·19-s − 7/5·25-s + 0.755·28-s + 2.87·31-s − 2.30·37-s − 0.609·43-s − 2/7·49-s − 0.277·52-s + 1.79·61-s + 7/8·64-s − 2.44·67-s + 1.63·73-s + 0.458·76-s − 0.450·79-s − 0.838·91-s + 0.406·97-s + 7/10·100-s + 1.57·103-s − 2.10·109-s + 1.13·112-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.47602425196733180808510235490, −6.97628079058080314939730537220, −6.80891826212390314326490246244, −6.35559472718337603287764484755, −6.24562481505812713463926433936, −6.06420997555128303906823363261, −5.27008504006369175030698491509, −5.21904369715254240316449956535, −4.67803989923594633740667786051, −4.36039868496485172908644907565, −3.89884842017188538670384766657, −3.68939238846573103703420248090, −3.27380712296299602714362207289, −2.85719586974089316958048605023, −2.44941844836081933712830884599, −1.99728663955535100055940092563, −1.47452825009956490658778358704, −0.834913515341981323344590190895, 0, 0,
0.834913515341981323344590190895, 1.47452825009956490658778358704, 1.99728663955535100055940092563, 2.44941844836081933712830884599, 2.85719586974089316958048605023, 3.27380712296299602714362207289, 3.68939238846573103703420248090, 3.89884842017188538670384766657, 4.36039868496485172908644907565, 4.67803989923594633740667786051, 5.21904369715254240316449956535, 5.27008504006369175030698491509, 6.06420997555128303906823363261, 6.24562481505812713463926433936, 6.35559472718337603287764484755, 6.80891826212390314326490246244, 6.97628079058080314939730537220, 7.47602425196733180808510235490
