| L(s) = 1 | + 2·3-s + 3·9-s − 2·11-s + 12·17-s − 6·25-s + 4·27-s − 4·33-s + 12·41-s + 16·43-s − 14·49-s + 24·51-s + 24·59-s − 8·67-s − 28·73-s − 12·75-s + 5·81-s + 24·83-s + 20·89-s − 28·97-s − 6·99-s − 8·107-s − 12·113-s + 3·121-s + 24·123-s + 127-s + 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.603·11-s + 2.91·17-s − 6/5·25-s + 0.769·27-s − 0.696·33-s + 1.87·41-s + 2.43·43-s − 2·49-s + 3.36·51-s + 3.12·59-s − 0.977·67-s − 3.27·73-s − 1.38·75-s + 5/9·81-s + 2.63·83-s + 2.11·89-s − 2.84·97-s − 0.603·99-s − 0.773·107-s − 1.12·113-s + 3/11·121-s + 2.16·123-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.698923945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.698923945\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.007653350633810471062865790375, −7.79883055392061183573611142640, −7.36587637640732761296700605801, −7.03448872626019280888548301992, −6.12991681965878756173970153874, −5.90293010636602553879430804715, −5.39498548145393328063580392099, −4.98634663126237155564427230653, −4.07317470493510196863296307169, −3.98296680233403436899260776008, −3.29088474012067366328421153205, −2.82255351943050054606485500714, −2.38765112618715460342688099346, −1.54699235007435022323925928448, −0.871830900111580766078276703461,
0.871830900111580766078276703461, 1.54699235007435022323925928448, 2.38765112618715460342688099346, 2.82255351943050054606485500714, 3.29088474012067366328421153205, 3.98296680233403436899260776008, 4.07317470493510196863296307169, 4.98634663126237155564427230653, 5.39498548145393328063580392099, 5.90293010636602553879430804715, 6.12991681965878756173970153874, 7.03448872626019280888548301992, 7.36587637640732761296700605801, 7.79883055392061183573611142640, 8.007653350633810471062865790375
