| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s + 2·19-s − 12·29-s + 8·31-s + 36-s + 20·41-s − 8·44-s + 14·49-s + 24·59-s − 4·61-s − 64-s + 16·71-s − 2·76-s + 8·79-s + 81-s − 20·89-s − 8·99-s + 20·101-s + 12·109-s + 12·116-s + 26·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 0.458·19-s − 2.22·29-s + 1.43·31-s + 1/6·36-s + 3.12·41-s − 1.20·44-s + 2·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 1.89·71-s − 0.229·76-s + 0.900·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s + 1.99·101-s + 1.14·109-s + 1.11·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.276603408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276603408\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.937411227781701973129571562601, −8.833965476724837117413008451687, −8.267413896694576318783735522270, −7.86037906367919305850001206604, −7.35752364706309258162195544685, −7.21758120764841977917678759360, −6.60228428996643886699393067599, −6.36182086690218974639245422670, −5.84528558062778653975451560080, −5.61592551294927645941695704551, −5.17264045444295940907439437345, −4.54228949037610847675954638480, −4.10370906250141825011030795705, −3.79421947765328079982345185482, −3.70210682105875246603949897418, −2.83315462286494875820042924471, −2.35682492344967270602662253752, −1.80072908839523862312797285160, −0.924079573645545502987278360893, −0.804953668745430946328674224290,
0.804953668745430946328674224290, 0.924079573645545502987278360893, 1.80072908839523862312797285160, 2.35682492344967270602662253752, 2.83315462286494875820042924471, 3.70210682105875246603949897418, 3.79421947765328079982345185482, 4.10370906250141825011030795705, 4.54228949037610847675954638480, 5.17264045444295940907439437345, 5.61592551294927645941695704551, 5.84528558062778653975451560080, 6.36182086690218974639245422670, 6.60228428996643886699393067599, 7.21758120764841977917678759360, 7.35752364706309258162195544685, 7.86037906367919305850001206604, 8.267413896694576318783735522270, 8.833965476724837117413008451687, 8.937411227781701973129571562601
