L(s) = 1 | + 6·9-s − 2·11-s − 4·19-s − 2·29-s + 4·31-s + 24·41-s − 49-s − 20·59-s + 8·61-s + 6·71-s − 18·79-s + 27·81-s + 12·89-s − 12·99-s + 24·101-s − 10·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
L(s) = 1 | + 2·9-s − 0.603·11-s − 0.917·19-s − 0.371·29-s + 0.718·31-s + 3.74·41-s − 1/7·49-s − 2.60·59-s + 1.02·61-s + 0.712·71-s − 2.02·79-s + 3·81-s + 1.27·89-s − 1.20·99-s + 2.38·101-s − 0.957·109-s − 1.72·121-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 + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.994071556\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.994071556\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.987580220894569765943456026911, −8.767676850893497549172386042600, −7.983756260835750712005208834580, −7.74662939867041886646242846993, −7.64626743882928180275023609940, −7.19080763430036140943586240738, −6.56093804928038261803541032505, −6.53115139993931420625363737273, −5.94224177464664418584149948307, −5.57794553737165345132575280987, −5.02738150763096456837183848231, −4.51084993773464118217159838039, −4.20888597714119995921158889225, −4.15525470121499606203466314814, −3.32876956629738001565344348671, −2.88356868792995965327967952381, −2.22956998807617659933941189387, −1.90652841912408102382557785599, −1.18700357345533808297043422356, −0.61307819382894208145769915765,
0.61307819382894208145769915765, 1.18700357345533808297043422356, 1.90652841912408102382557785599, 2.22956998807617659933941189387, 2.88356868792995965327967952381, 3.32876956629738001565344348671, 4.15525470121499606203466314814, 4.20888597714119995921158889225, 4.51084993773464118217159838039, 5.02738150763096456837183848231, 5.57794553737165345132575280987, 5.94224177464664418584149948307, 6.53115139993931420625363737273, 6.56093804928038261803541032505, 7.19080763430036140943586240738, 7.64626743882928180275023609940, 7.74662939867041886646242846993, 7.983756260835750712005208834580, 8.767676850893497549172386042600, 8.987580220894569765943456026911