L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s + 4·15-s + 2·17-s + 2·21-s − 6·25-s + 4·27-s + 2·35-s − 8·37-s + 10·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 4·51-s − 4·59-s − 63-s + 4·67-s + 12·75-s − 4·79-s − 11·81-s + 20·83-s − 4·85-s − 22·89-s − 2·101-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.03·15-s + 0.485·17-s + 0.436·21-s − 6/5·25-s + 0.769·27-s + 0.338·35-s − 1.31·37-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 0.520·59-s − 0.125·63-s + 0.488·67-s + 1.38·75-s − 0.450·79-s − 1.22·81-s + 2.19·83-s − 0.433·85-s − 2.33·89-s − 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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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
−8.006810999334306246014221316282, −7.41679787328512967221220736949, −7.26833237112847662899203072491, −6.57889005868593029480881427029, −6.25217444268487149618739817919, −5.66400557363162770922137856429, −5.47422062256893143755282411218, −4.84514450356068525065576764403, −4.27252669055871372625228478133, −3.86681736268164082609446233755, −3.32746840802018450087510866921, −2.67603562035073970243533236266, −1.83169799468905694838870970035, −0.833309481322196499491609488437, 0,
0.833309481322196499491609488437, 1.83169799468905694838870970035, 2.67603562035073970243533236266, 3.32746840802018450087510866921, 3.86681736268164082609446233755, 4.27252669055871372625228478133, 4.84514450356068525065576764403, 5.47422062256893143755282411218, 5.66400557363162770922137856429, 6.25217444268487149618739817919, 6.57889005868593029480881427029, 7.26833237112847662899203072491, 7.41679787328512967221220736949, 8.006810999334306246014221316282