L(s) = 1 | + 2-s − 4-s + 4·5-s − 3·8-s − 3·9-s + 4·10-s − 16-s − 4·17-s − 3·18-s − 4·20-s + 3·25-s + 14·29-s + 5·32-s − 4·34-s + 3·36-s − 4·37-s − 12·40-s − 12·41-s − 12·45-s + 49-s + 3·50-s − 2·53-s + 14·58-s + 4·61-s + 7·64-s + 4·68-s + 9·72-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s − 9-s + 1.26·10-s − 1/4·16-s − 0.970·17-s − 0.707·18-s − 0.894·20-s + 3/5·25-s + 2.59·29-s + 0.883·32-s − 0.685·34-s + 1/2·36-s − 0.657·37-s − 1.89·40-s − 1.87·41-s − 1.78·45-s + 1/7·49-s + 0.424·50-s − 0.274·53-s + 1.83·58-s + 0.512·61-s + 7/8·64-s + 0.485·68-s + 1.06·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.912557315676358567794666403783, −7.07252118152093106605624065275, −6.60200851238385647996145024794, −6.31951729401620381379585261696, −6.02336898838701487042227128106, −5.45291163293155166850096601088, −5.13548703344812413123160320253, −4.82405407040722146505324502341, −4.19728318460680687523126437985, −3.59568732569069429908453888433, −2.98071006984466882116716931515, −2.54351357308048197224785892344, −2.05715121046755611720399459632, −1.21189548615431144984756203024, 0,
1.21189548615431144984756203024, 2.05715121046755611720399459632, 2.54351357308048197224785892344, 2.98071006984466882116716931515, 3.59568732569069429908453888433, 4.19728318460680687523126437985, 4.82405407040722146505324502341, 5.13548703344812413123160320253, 5.45291163293155166850096601088, 6.02336898838701487042227128106, 6.31951729401620381379585261696, 6.60200851238385647996145024794, 7.07252118152093106605624065275, 7.912557315676358567794666403783