L(s) = 1 | − 2·5-s + 8·11-s + 8·19-s − 25-s − 4·29-s − 4·41-s + 10·49-s − 16·55-s − 24·59-s + 20·61-s + 16·71-s + 32·79-s + 12·89-s − 16·95-s + 12·101-s − 12·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 2.41·11-s + 1.83·19-s − 1/5·25-s − 0.742·29-s − 0.624·41-s + 10/7·49-s − 2.15·55-s − 3.12·59-s + 2.56·61-s + 1.89·71-s + 3.60·79-s + 1.27·89-s − 1.64·95-s + 1.19·101-s − 1.14·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.859087169\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.859087169\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.085037475724452118744105110761, −8.618600558941157381538235854594, −8.047787842305659490877945318335, −7.958656823230094324663539445972, −7.28906784590888523379442777290, −7.22814937150268353410915063962, −6.75761875063465332873077022641, −6.24845125368905720309311453147, −6.10003862581920162839204184649, −5.48415578537040930576062345674, −4.89599409998695277558085175436, −4.83913664025364025434978902763, −3.88019536285413255719020019591, −3.84662858531755992264644307710, −3.59738082068896792780058195560, −3.06374746357818533831926640351, −2.27690564688485949734271957388, −1.74376284555916480169795414151, −1.09295563200955017277706632144, −0.64827334485885844710405927483,
0.64827334485885844710405927483, 1.09295563200955017277706632144, 1.74376284555916480169795414151, 2.27690564688485949734271957388, 3.06374746357818533831926640351, 3.59738082068896792780058195560, 3.84662858531755992264644307710, 3.88019536285413255719020019591, 4.83913664025364025434978902763, 4.89599409998695277558085175436, 5.48415578537040930576062345674, 6.10003862581920162839204184649, 6.24845125368905720309311453147, 6.75761875063465332873077022641, 7.22814937150268353410915063962, 7.28906784590888523379442777290, 7.958656823230094324663539445972, 8.047787842305659490877945318335, 8.618600558941157381538235854594, 9.085037475724452118744105110761