L(s) = 1 | − 2·3-s + 3·9-s − 4·17-s − 6·25-s − 4·27-s − 12·29-s − 8·43-s + 10·49-s + 8·51-s − 20·53-s − 28·61-s + 12·75-s − 32·79-s + 5·81-s + 24·87-s − 20·101-s + 16·103-s + 24·107-s + 12·113-s + 6·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 20·147-s + 149-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 0.970·17-s − 6/5·25-s − 0.769·27-s − 2.22·29-s − 1.21·43-s + 10/7·49-s + 1.12·51-s − 2.74·53-s − 3.58·61-s + 1.38·75-s − 3.60·79-s + 5/9·81-s + 2.57·87-s − 1.99·101-s + 1.57·103-s + 2.32·107-s + 1.12·113-s + 6/11·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.868846233846508506388452521795, −8.813732844963216489184794391429, −7.935730992416060941358040400705, −7.80433978468021520564411993845, −7.21570051869570027856824075318, −7.12695148958217466605537996887, −6.47007881210334126460844651450, −5.98465857057036710103781826432, −5.92532997716241188678680318446, −5.48453599620452708294071401486, −4.79193386425839096467955534518, −4.58728859259556129333279487065, −4.17737100000295686817537665077, −3.54236846157479395782221387964, −3.17816743860810052265591214063, −2.35866999897939738587899993063, −1.72379456102557725178058192806, −1.42242276437796690933912837265, 0, 0,
1.42242276437796690933912837265, 1.72379456102557725178058192806, 2.35866999897939738587899993063, 3.17816743860810052265591214063, 3.54236846157479395782221387964, 4.17737100000295686817537665077, 4.58728859259556129333279487065, 4.79193386425839096467955534518, 5.48453599620452708294071401486, 5.92532997716241188678680318446, 5.98465857057036710103781826432, 6.47007881210334126460844651450, 7.12695148958217466605537996887, 7.21570051869570027856824075318, 7.80433978468021520564411993845, 7.935730992416060941358040400705, 8.813732844963216489184794391429, 8.868846233846508506388452521795