L(s) = 1 | + 7-s − 16·23-s + 6·25-s − 4·29-s − 12·37-s + 16·43-s + 49-s + 20·53-s − 24·67-s − 16·79-s + 24·107-s + 20·109-s − 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + 167-s − 26·169-s + 173-s + 6·175-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 3.33·23-s + 6/5·25-s − 0.742·29-s − 1.97·37-s + 2.43·43-s + 1/7·49-s + 2.74·53-s − 2.93·67-s − 1.80·79-s + 2.32·107-s + 1.91·109-s − 1.12·113-s − 2·121-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 + 0.0798·157-s − 1.26·161-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.453·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.611574819\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611574819\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.69639740408856738387093436756, −7.36849069017085576948583776260, −7.18786399959710400886593887034, −6.46556088998900741811578342958, −6.03463616838718144143011248354, −5.56463127308816764910872719867, −5.49102837407369040609817160877, −4.54986168178647908951568994162, −4.27897299263841475745331538633, −3.89113776702465302721269609787, −3.28822043116364534811243543739, −2.61365926232435527180870628710, −2.03505353231784433825131603728, −1.58802930336018620100956346509, −0.51874703726271079374177335007,
0.51874703726271079374177335007, 1.58802930336018620100956346509, 2.03505353231784433825131603728, 2.61365926232435527180870628710, 3.28822043116364534811243543739, 3.89113776702465302721269609787, 4.27897299263841475745331538633, 4.54986168178647908951568994162, 5.49102837407369040609817160877, 5.56463127308816764910872719867, 6.03463616838718144143011248354, 6.46556088998900741811578342958, 7.18786399959710400886593887034, 7.36849069017085576948583776260, 7.69639740408856738387093436756