L(s) = 1 | − 4-s + 6·11-s + 16-s + 8·19-s + 18·29-s + 10·31-s − 6·44-s − 11·49-s − 18·59-s − 2·61-s − 64-s − 8·76-s + 32·79-s − 12·89-s + 6·101-s − 4·109-s − 18·116-s + 5·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.80·11-s + 1/4·16-s + 1.83·19-s + 3.34·29-s + 1.79·31-s − 0.904·44-s − 1.57·49-s − 2.34·59-s − 0.256·61-s − 1/8·64-s − 0.917·76-s + 3.60·79-s − 1.27·89-s + 0.597·101-s − 0.383·109-s − 1.67·116-s + 5/11·121-s − 0.898·124-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 + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.111264328\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.111264328\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.326125621414466357425643526662, −7.989441585400118982009954962358, −7.64858056663539622694664322680, −7.22709627955859053005018759586, −6.63345400432238205142595429399, −6.53714679799580387883892891291, −6.27942319834309992364544686522, −5.95603111496467144347020744148, −5.15056508179130536814559404148, −5.06810941158377703294819402288, −4.64487834160716412977964386495, −4.30745877570547816007435966240, −3.95501478890688052316325447945, −3.34386277593058466746112364740, −2.95424717823972850254290737686, −2.92203061884521780410050798622, −2.00586297407569526267436232887, −1.39021069772886326040703868159, −1.03613172181913335854310145249, −0.65787606373635275209591632574,
0.65787606373635275209591632574, 1.03613172181913335854310145249, 1.39021069772886326040703868159, 2.00586297407569526267436232887, 2.92203061884521780410050798622, 2.95424717823972850254290737686, 3.34386277593058466746112364740, 3.95501478890688052316325447945, 4.30745877570547816007435966240, 4.64487834160716412977964386495, 5.06810941158377703294819402288, 5.15056508179130536814559404148, 5.95603111496467144347020744148, 6.27942319834309992364544686522, 6.53714679799580387883892891291, 6.63345400432238205142595429399, 7.22709627955859053005018759586, 7.64858056663539622694664322680, 7.989441585400118982009954962358, 8.326125621414466357425643526662