| L(s) = 1 | − 4-s − 3·5-s + 2·8-s − 2·9-s + 4·11-s + 4·13-s + 16-s + 6·19-s + 3·20-s − 2·23-s + 8·25-s − 2·29-s − 4·32-s + 2·36-s − 6·40-s − 4·44-s + 6·45-s + 16·47-s + 12·49-s − 4·52-s − 12·55-s + 3·64-s − 12·65-s − 4·72-s − 6·76-s − 12·79-s − 3·80-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.34·5-s + 0.707·8-s − 2/3·9-s + 1.20·11-s + 1.10·13-s + 1/4·16-s + 1.37·19-s + 0.670·20-s − 0.417·23-s + 8/5·25-s − 0.371·29-s − 0.707·32-s + 1/3·36-s − 0.948·40-s − 0.603·44-s + 0.894·45-s + 2.33·47-s + 12/7·49-s − 0.554·52-s − 1.61·55-s + 3/8·64-s − 1.48·65-s − 0.471·72-s − 0.688·76-s − 1.35·79-s − 0.335·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503771458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503771458\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
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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
−8.676938213170042432854403055351, −8.236346967153250731616112505921, −7.59362614184224631823320460217, −7.31276940731260751352496463100, −7.00425316097302373219286218836, −6.19410263438991494825763717770, −5.77758701424140404195022633832, −5.31166878420092529722720124802, −4.60170209483041450565863246868, −4.10873251579165287804507244955, −3.77692180581513809556450353478, −3.35707750384324760132968956490, −2.55545007658910070915998436494, −1.39494847502604546798575569456, −0.76387878351247798619711675052,
0.76387878351247798619711675052, 1.39494847502604546798575569456, 2.55545007658910070915998436494, 3.35707750384324760132968956490, 3.77692180581513809556450353478, 4.10873251579165287804507244955, 4.60170209483041450565863246868, 5.31166878420092529722720124802, 5.77758701424140404195022633832, 6.19410263438991494825763717770, 7.00425316097302373219286218836, 7.31276940731260751352496463100, 7.59362614184224631823320460217, 8.236346967153250731616112505921, 8.676938213170042432854403055351
