| L(s) = 1 | + 4·5-s − 4·13-s + 4·17-s + 2·25-s − 20·37-s + 8·41-s + 28·53-s − 24·61-s − 16·65-s − 28·73-s + 18·81-s + 16·85-s − 28·97-s + 40·101-s + 36·113-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.10·13-s + 0.970·17-s + 2/5·25-s − 3.28·37-s + 1.24·41-s + 3.84·53-s − 3.07·61-s − 1.98·65-s − 3.27·73-s + 2·81-s + 1.73·85-s − 2.84·97-s + 3.98·101-s + 3.38·113-s + 1.81·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 542 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3326 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.520030397748944726355109838589, −8.242490332142431549097990350949, −7.905255167452555018665217257026, −7.66452995186847811374692307695, −7.40134027715321106397179914974, −7.06817466296699457556799177892, −6.99415616841921703025163970581, −6.82308850182998501342410649284, −6.20733675697868705625716078948, −6.00283143055561027617739772875, −5.85169404000169724182152534028, −5.51534163991162366790102186721, −5.44971124627777014870701033416, −5.20647333192485474939450409220, −4.56981473109667500251945808067, −4.39256177429860271760739604785, −4.37068983109001058438863996460, −3.51405606391579631897399383262, −3.31913861834550304979634410517, −3.17273449765967984060066023065, −2.46614160490971712677917911034, −2.27341516007885997535894760599, −1.75002397272155427007061779895, −1.66652469312998831091684148606, −0.67909613830254790842068001412,
0.67909613830254790842068001412, 1.66652469312998831091684148606, 1.75002397272155427007061779895, 2.27341516007885997535894760599, 2.46614160490971712677917911034, 3.17273449765967984060066023065, 3.31913861834550304979634410517, 3.51405606391579631897399383262, 4.37068983109001058438863996460, 4.39256177429860271760739604785, 4.56981473109667500251945808067, 5.20647333192485474939450409220, 5.44971124627777014870701033416, 5.51534163991162366790102186721, 5.85169404000169724182152534028, 6.00283143055561027617739772875, 6.20733675697868705625716078948, 6.82308850182998501342410649284, 6.99415616841921703025163970581, 7.06817466296699457556799177892, 7.40134027715321106397179914974, 7.66452995186847811374692307695, 7.905255167452555018665217257026, 8.242490332142431549097990350949, 8.520030397748944726355109838589
