| L(s) = 1 | + 2·5-s − 8·7-s + 4·9-s + 6·11-s − 4·19-s + 3·25-s − 16·35-s + 20·37-s − 16·43-s + 8·45-s + 34·49-s − 12·53-s + 12·55-s − 32·63-s − 48·77-s + 4·79-s + 7·81-s − 24·83-s − 8·95-s + 4·97-s + 24·99-s − 36·113-s + 25·121-s + 4·125-s + 127-s + 131-s + 32·133-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.02·7-s + 4/3·9-s + 1.80·11-s − 0.917·19-s + 3/5·25-s − 2.70·35-s + 3.28·37-s − 2.43·43-s + 1.19·45-s + 34/7·49-s − 1.64·53-s + 1.61·55-s − 4.03·63-s − 5.47·77-s + 0.450·79-s + 7/9·81-s − 2.63·83-s − 0.820·95-s + 0.406·97-s + 2.41·99-s − 3.38·113-s + 2.27·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678736243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678736243\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.128663594818806879450882406006, −8.485925820643422940386286056022, −8.003221123283887914541766775712, −7.56651941089178150800132656085, −6.94448579817377048715639098434, −6.76330715335996580324673466526, −6.49617894075015820168505406100, −6.29887937457082838703022484141, −6.07370892866409056681443658935, −5.54703408153792384400815134388, −4.91273086489893400691573447556, −4.26478816004000506844697515950, −4.20802357661074006711521015741, −3.60117952680927010723954107335, −3.37766797588235592046942317033, −2.62774291852375960533648519291, −2.57278805372048431934767436617, −1.43957727227369303982034111407, −1.41208636245298268140110172989, −0.39637858679969621621876243927,
0.39637858679969621621876243927, 1.41208636245298268140110172989, 1.43957727227369303982034111407, 2.57278805372048431934767436617, 2.62774291852375960533648519291, 3.37766797588235592046942317033, 3.60117952680927010723954107335, 4.20802357661074006711521015741, 4.26478816004000506844697515950, 4.91273086489893400691573447556, 5.54703408153792384400815134388, 6.07370892866409056681443658935, 6.29887937457082838703022484141, 6.49617894075015820168505406100, 6.76330715335996580324673466526, 6.94448579817377048715639098434, 7.56651941089178150800132656085, 8.003221123283887914541766775712, 8.485925820643422940386286056022, 9.128663594818806879450882406006
