L(s) = 1 | − 2·3-s + 3·9-s + 6·11-s − 6·17-s + 2·19-s − 10·27-s − 12·33-s − 6·41-s − 20·43-s − 14·49-s + 12·51-s − 4·57-s − 12·59-s + 14·67-s − 2·73-s + 20·81-s − 18·83-s + 18·89-s − 20·97-s + 18·99-s − 6·107-s + 18·113-s + 11·121-s + 12·123-s + 127-s + 40·129-s + 131-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 1.80·11-s − 1.45·17-s + 0.458·19-s − 1.92·27-s − 2.08·33-s − 0.937·41-s − 3.04·43-s − 2·49-s + 1.68·51-s − 0.529·57-s − 1.56·59-s + 1.71·67-s − 0.234·73-s + 20/9·81-s − 1.97·83-s + 1.90·89-s − 2.03·97-s + 1.80·99-s − 0.580·107-s + 1.69·113-s + 121-s + 1.08·123-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.88060018305853463525442801332, −7.31174055608281766025293041415, −6.90485898506978115661051188541, −6.75415847895087287747686132585, −6.50662421180971289549145324764, −6.18205069184680994509084516198, −5.77674297318912648955856560663, −5.37644926878243155036929436780, −4.86340413548539080120202098662, −4.72793801947666842607778454898, −4.34490731685373620894884309255, −3.78442074354798081244950755146, −3.52020937726253794948522753565, −3.20671257835573326590524310896, −2.38742513462064796930137608277, −1.84666064780823217793452221722, −1.51453439629908101213765663735, −1.13952606903357435585572887555, 0, 0,
1.13952606903357435585572887555, 1.51453439629908101213765663735, 1.84666064780823217793452221722, 2.38742513462064796930137608277, 3.20671257835573326590524310896, 3.52020937726253794948522753565, 3.78442074354798081244950755146, 4.34490731685373620894884309255, 4.72793801947666842607778454898, 4.86340413548539080120202098662, 5.37644926878243155036929436780, 5.77674297318912648955856560663, 6.18205069184680994509084516198, 6.50662421180971289549145324764, 6.75415847895087287747686132585, 6.90485898506978115661051188541, 7.31174055608281766025293041415, 7.88060018305853463525442801332