| L(s) = 1 | + 6·5-s − 12·11-s − 164·13-s − 30·17-s − 68·19-s + 216·23-s + 125·25-s − 492·29-s + 112·31-s − 110·37-s + 492·41-s − 344·43-s + 192·47-s + 558·53-s − 72·55-s + 540·59-s − 110·61-s − 984·65-s − 140·67-s + 1.68e3·71-s + 550·73-s + 208·79-s − 1.03e3·83-s − 180·85-s − 1.39e3·89-s − 408·95-s + 3.17e3·97-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 0.328·11-s − 3.49·13-s − 0.428·17-s − 0.821·19-s + 1.95·23-s + 25-s − 3.15·29-s + 0.648·31-s − 0.488·37-s + 1.87·41-s − 1.21·43-s + 0.595·47-s + 1.44·53-s − 0.176·55-s + 1.19·59-s − 0.230·61-s − 1.87·65-s − 0.255·67-s + 2.80·71-s + 0.881·73-s + 0.296·79-s − 1.36·83-s − 0.229·85-s − 1.66·89-s − 0.440·95-s + 3.32·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.706564106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706564106\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 30 T - 4013 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 68 T - 2235 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 216 T + 34489 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 246 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 112 T - 17247 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 192 T - 66959 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 558 T + 162487 T^{2} - 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 540 T + 86221 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 110 T - 214881 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 140 T - 281163 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 550 T - 86517 T^{2} - 550 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 208 T - 449775 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 516 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1398 T + 1249435 T^{2} + 1398 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1586 T + p^{3} T^{2} )^{2} \) |
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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
−9.052221382273805301408406346154, −9.020422165924869893164370914615, −8.375699390839973244842821999413, −7.82018216569863322577176746571, −7.37959852258347167941793365177, −7.26146181736350600957438556331, −6.81063842696461118897875355416, −6.56478314621434743514068378396, −5.66017500480244854917819616112, −5.46909049364320176268653737552, −5.01126581062462448500396139857, −4.83991701422585894997080783907, −4.23005774667191642212710346574, −3.74940038934871434895651144395, −2.93537256464650623688272063301, −2.64690620064750497691621759005, −2.12033042530444823069714710582, −1.94108027749357494432653834038, −0.802563063807657712302807523489, −0.34269278439099835618194304101,
0.34269278439099835618194304101, 0.802563063807657712302807523489, 1.94108027749357494432653834038, 2.12033042530444823069714710582, 2.64690620064750497691621759005, 2.93537256464650623688272063301, 3.74940038934871434895651144395, 4.23005774667191642212710346574, 4.83991701422585894997080783907, 5.01126581062462448500396139857, 5.46909049364320176268653737552, 5.66017500480244854917819616112, 6.56478314621434743514068378396, 6.81063842696461118897875355416, 7.26146181736350600957438556331, 7.37959852258347167941793365177, 7.82018216569863322577176746571, 8.375699390839973244842821999413, 9.020422165924869893164370914615, 9.052221382273805301408406346154
