L(s) = 1 | − 4·9-s + 2·13-s − 6·17-s − 5·25-s + 8·29-s − 6·37-s − 8·41-s − 8·49-s − 2·53-s − 12·61-s + 6·73-s + 7·81-s − 10·97-s + 4·101-s − 12·109-s − 2·113-s − 8·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 4/3·9-s + 0.554·13-s − 1.45·17-s − 25-s + 1.48·29-s − 0.986·37-s − 1.24·41-s − 8/7·49-s − 0.274·53-s − 1.53·61-s + 0.702·73-s + 7/9·81-s − 1.01·97-s + 0.398·101-s − 1.14·109-s − 0.188·113-s − 0.739·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.246293219893359048315475450834, −8.726846220813086816751117453863, −8.368022551806504658657706821578, −8.056070206846718192514137029569, −7.29673004418918807387745827221, −6.56993334744099219984299302189, −6.38406773230374130971567808287, −5.76214078779174672208980511472, −5.10506438034023107431865781770, −4.62718799555261247340452158378, −3.84829219974676613239566862246, −3.18666727785042166214265809215, −2.54428325685487940670603121172, −1.65410814104439059357742212771, 0,
1.65410814104439059357742212771, 2.54428325685487940670603121172, 3.18666727785042166214265809215, 3.84829219974676613239566862246, 4.62718799555261247340452158378, 5.10506438034023107431865781770, 5.76214078779174672208980511472, 6.38406773230374130971567808287, 6.56993334744099219984299302189, 7.29673004418918807387745827221, 8.056070206846718192514137029569, 8.368022551806504658657706821578, 8.726846220813086816751117453863, 9.246293219893359048315475450834