| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s + 12·13-s + 16-s + 4·17-s + 18-s + 2·20-s + 3·25-s + 12·26-s − 20·29-s + 32-s + 4·34-s + 36-s + 12·37-s + 2·40-s + 4·41-s + 2·45-s − 14·49-s + 3·50-s + 12·52-s − 20·53-s − 20·58-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 3.32·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s + 2.35·26-s − 3.71·29-s + 0.176·32-s + 0.685·34-s + 1/6·36-s + 1.97·37-s + 0.316·40-s + 0.624·41-s + 0.298·45-s − 2·49-s + 0.424·50-s + 1.66·52-s − 2.74·53-s − 2.62·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 871200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 871200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.159444632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.159444632\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.029851509254933986565522519350, −7.57573281855054290442967607898, −7.57220889964285579933989784409, −6.48861488882352966289362258399, −6.26970322751556902553424782587, −5.88065102738235433506263724139, −5.81872055345238569065006562950, −5.01691179270026500847515723163, −4.54831431199383754044494926069, −3.81607349420999372453813357157, −3.43907663083633253170978783027, −3.26184904194178307659961725048, −2.09507854443047709409494683998, −1.62793361398186118811552584678, −1.09509658039844740721467637834,
1.09509658039844740721467637834, 1.62793361398186118811552584678, 2.09507854443047709409494683998, 3.26184904194178307659961725048, 3.43907663083633253170978783027, 3.81607349420999372453813357157, 4.54831431199383754044494926069, 5.01691179270026500847515723163, 5.81872055345238569065006562950, 5.88065102738235433506263724139, 6.26970322751556902553424782587, 6.48861488882352966289362258399, 7.57220889964285579933989784409, 7.57573281855054290442967607898, 8.029851509254933986565522519350
