| L(s) = 1 | − 5-s − 9-s + 13-s + 2·17-s − 4·25-s + 29-s − 7·37-s + 10·41-s + 45-s + 5·49-s − 14·53-s − 11·61-s − 65-s + 12·73-s + 81-s − 2·85-s − 5·89-s + 20·97-s − 11·101-s − 16·109-s + 2·113-s − 117-s − 20·121-s + 9·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1/3·9-s + 0.277·13-s + 0.485·17-s − 4/5·25-s + 0.185·29-s − 1.15·37-s + 1.56·41-s + 0.149·45-s + 5/7·49-s − 1.92·53-s − 1.40·61-s − 0.124·65-s + 1.40·73-s + 1/9·81-s − 0.216·85-s − 0.529·89-s + 2.03·97-s − 1.09·101-s − 1.53·109-s + 0.188·113-s − 0.0924·117-s − 1.81·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
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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
−7.931172082925463185809780147812, −7.63623029352616131769050619633, −7.15043841723428652801821628125, −6.52426926220127212728003020234, −6.23892264386439664477638266297, −5.68517291286598822892829949008, −5.27657809093124687000445066506, −4.73505467051198312338187270190, −4.17051500539146157820923983347, −3.72227413582024811934999571076, −3.20199322402935547852744448652, −2.63494012053467085835293090044, −1.89600042290926202807506135336, −1.11383627519008104269331000504, 0,
1.11383627519008104269331000504, 1.89600042290926202807506135336, 2.63494012053467085835293090044, 3.20199322402935547852744448652, 3.72227413582024811934999571076, 4.17051500539146157820923983347, 4.73505467051198312338187270190, 5.27657809093124687000445066506, 5.68517291286598822892829949008, 6.23892264386439664477638266297, 6.52426926220127212728003020234, 7.15043841723428652801821628125, 7.63623029352616131769050619633, 7.931172082925463185809780147812
