| L(s) = 1 | + 3·4-s − 9-s − 8·11-s + 5·16-s + 8·19-s − 2·29-s − 16·31-s − 3·36-s − 12·41-s − 24·44-s − 2·49-s + 24·59-s − 20·61-s + 3·64-s − 16·71-s + 24·76-s + 81-s + 12·89-s + 8·99-s − 4·101-s − 28·109-s − 6·116-s + 26·121-s − 48·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.371·29-s − 2.87·31-s − 1/2·36-s − 1.87·41-s − 3.61·44-s − 2/7·49-s + 3.12·59-s − 2.56·61-s + 3/8·64-s − 1.89·71-s + 2.75·76-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 0.398·101-s − 2.68·109-s − 0.557·116-s + 2.36·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.738032224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738032224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.070889185073528998810106383266, −9.057313080397504636498094233117, −8.309521536089128087636987815792, −7.83926668980204009205845404972, −7.78878616378449729901467018796, −7.21947970859282073331256958176, −7.15767418905591412524372299714, −6.65187256612330474512955645408, −6.06387341895018438278700109296, −5.52295313792686514588730913045, −5.31534244219314263306376428164, −5.30132582455757463426531845302, −4.49913024472449427293520856697, −3.70715960809490838707105568047, −3.31328980230958298925207899687, −2.87508803831621090236192534290, −2.61956463322222460130096929243, −1.79621527603148595073834414844, −1.68432752889768692649237709781, −0.41368165828808100915162555769,
0.41368165828808100915162555769, 1.68432752889768692649237709781, 1.79621527603148595073834414844, 2.61956463322222460130096929243, 2.87508803831621090236192534290, 3.31328980230958298925207899687, 3.70715960809490838707105568047, 4.49913024472449427293520856697, 5.30132582455757463426531845302, 5.31534244219314263306376428164, 5.52295313792686514588730913045, 6.06387341895018438278700109296, 6.65187256612330474512955645408, 7.15767418905591412524372299714, 7.21947970859282073331256958176, 7.78878616378449729901467018796, 7.83926668980204009205845404972, 8.309521536089128087636987815792, 9.057313080397504636498094233117, 9.070889185073528998810106383266
