| L(s) = 1 | − 9·9-s + 28·11-s − 38·19-s − 28·29-s + 266·31-s + 168·41-s + 661·49-s + 388·59-s − 34·61-s + 1.65e3·71-s + 1.10e3·79-s + 81·81-s + 2.20e3·89-s − 252·99-s + 1.10e3·101-s + 3.68e3·109-s − 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.767·11-s − 0.458·19-s − 0.179·29-s + 1.54·31-s + 0.639·41-s + 1.92·49-s + 0.856·59-s − 0.0713·61-s + 2.76·71-s + 1.57·79-s + 1/9·81-s + 2.62·89-s − 0.255·99-s + 1.08·101-s + 3.23·109-s − 1.55·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.141106006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.141106006\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 661 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7710 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 133 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34742 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 131125 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 39546 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 89818 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 194 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 175117 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 828 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 453134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 552 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1123410 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1104 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1118065 T^{2} + p^{6} T^{4} \) |
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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
−10.33632274258952882546110309110, −10.22257743998145646090133329610, −9.388110656755244807452141179795, −9.342480210181074280669813404427, −8.612421736483666771433280564708, −8.478318440346804837615389793335, −7.73148481151146225187159570789, −7.52872087305653300171032780579, −6.64919088844804273824691245604, −6.58902989762741280644121028035, −5.96241722043928980122202333366, −5.52111305824844172373223341601, −4.78930129485965841094195935731, −4.50809847561764285088880805692, −3.60669195043565804127643780183, −3.55071016051753500917026201096, −2.32168395748992946324983042550, −2.29904763295894959878539969108, −1.06905548052865161491382716048, −0.61610103676195062383139678494,
0.61610103676195062383139678494, 1.06905548052865161491382716048, 2.29904763295894959878539969108, 2.32168395748992946324983042550, 3.55071016051753500917026201096, 3.60669195043565804127643780183, 4.50809847561764285088880805692, 4.78930129485965841094195935731, 5.52111305824844172373223341601, 5.96241722043928980122202333366, 6.58902989762741280644121028035, 6.64919088844804273824691245604, 7.52872087305653300171032780579, 7.73148481151146225187159570789, 8.478318440346804837615389793335, 8.612421736483666771433280564708, 9.342480210181074280669813404427, 9.388110656755244807452141179795, 10.22257743998145646090133329610, 10.33632274258952882546110309110
