| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s + 4·17-s − 16·23-s − 25-s + 4·27-s + 4·29-s − 3·36-s − 24·43-s + 2·48-s − 2·49-s + 8·51-s + 20·53-s − 20·61-s − 64-s − 4·68-s − 32·69-s − 2·75-s − 16·79-s + 5·81-s + 8·87-s + 16·92-s + 100-s − 20·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s + 0.970·17-s − 3.33·23-s − 1/5·25-s + 0.769·27-s + 0.742·29-s − 1/2·36-s − 3.65·43-s + 0.288·48-s − 2/7·49-s + 1.12·51-s + 2.74·53-s − 2.56·61-s − 1/8·64-s − 0.485·68-s − 3.85·69-s − 0.230·75-s − 1.80·79-s + 5/9·81-s + 0.857·87-s + 1.66·92-s + 1/10·100-s − 1.99·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525864898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525864898\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.454568999041999285548090878256, −8.156534110156454528291463361006, −7.902779101384107378741333793771, −7.38974578229457086264478775725, −7.16404990992205796268208567653, −6.55477693633447694789149361466, −6.36681463879856556672251198774, −5.81744961166176221543510098640, −5.50535974498790619140837492969, −5.16922959109749358835537838779, −4.46948761712770069045514210303, −4.31199544402349728218557219639, −3.93742026584893608206283595605, −3.41461401347423865146825602999, −3.23796013966328258236053777514, −2.64268412581824663025404423360, −2.16965592687440240716886965186, −1.61429504050987896324182304044, −1.37644351106380768032526982252, −0.29520426968162943152674472081,
0.29520426968162943152674472081, 1.37644351106380768032526982252, 1.61429504050987896324182304044, 2.16965592687440240716886965186, 2.64268412581824663025404423360, 3.23796013966328258236053777514, 3.41461401347423865146825602999, 3.93742026584893608206283595605, 4.31199544402349728218557219639, 4.46948761712770069045514210303, 5.16922959109749358835537838779, 5.50535974498790619140837492969, 5.81744961166176221543510098640, 6.36681463879856556672251198774, 6.55477693633447694789149361466, 7.16404990992205796268208567653, 7.38974578229457086264478775725, 7.902779101384107378741333793771, 8.156534110156454528291463361006, 8.454568999041999285548090878256
