| L(s) = 1 | − 2-s − 3-s + 6-s + 3·7-s − 13-s − 3·14-s − 17-s − 19-s − 3·21-s − 23-s + 3·25-s + 26-s + 34-s − 37-s + 38-s + 39-s + 3·41-s + 3·42-s − 43-s + 46-s − 47-s + 6·49-s − 3·50-s + 51-s + 57-s + 69-s + 74-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 3·7-s − 13-s − 3·14-s − 17-s − 19-s − 3·21-s − 23-s + 3·25-s + 26-s + 34-s − 37-s + 38-s + 39-s + 3·41-s + 3·42-s − 43-s + 46-s − 47-s + 6·49-s − 3·50-s + 51-s + 57-s + 69-s + 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23639903 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23639903 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2130119663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2130119663\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91187458018416015170698685336, −10.78250559542412005593668909887, −10.25481607468148037229229356366, −9.864391262451580615867045765893, −9.528162202367427769248088987351, −8.986941469572290249075414720670, −8.879016758633469552542568523203, −8.460803414283570476536541005022, −8.407920308474781358938964039201, −8.040717908667976107861741553089, −7.57176915215757514066040403834, −7.28825108904406493133329166840, −6.91591547070030771755669009401, −6.59262895565668750672194969708, −5.95907407986090641156811679654, −5.75398581107553119917939330072, −5.14138460047647613929986935445, −4.90714365739911539367664818202, −4.80172147591202763029069042981, −4.17665141758212569550545438696, −4.05970639873068577016350463516, −2.86385095983606244584232727242, −2.41618319312512938855520846450, −1.89021135447327534675760110407, −1.19299410512603630683171186646,
1.19299410512603630683171186646, 1.89021135447327534675760110407, 2.41618319312512938855520846450, 2.86385095983606244584232727242, 4.05970639873068577016350463516, 4.17665141758212569550545438696, 4.80172147591202763029069042981, 4.90714365739911539367664818202, 5.14138460047647613929986935445, 5.75398581107553119917939330072, 5.95907407986090641156811679654, 6.59262895565668750672194969708, 6.91591547070030771755669009401, 7.28825108904406493133329166840, 7.57176915215757514066040403834, 8.040717908667976107861741553089, 8.407920308474781358938964039201, 8.460803414283570476536541005022, 8.879016758633469552542568523203, 8.986941469572290249075414720670, 9.528162202367427769248088987351, 9.864391262451580615867045765893, 10.25481607468148037229229356366, 10.78250559542412005593668909887, 10.91187458018416015170698685336
