| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 3·9-s + 8·11-s − 2·12-s − 16-s + 17-s − 3·18-s − 8·19-s − 8·22-s + 6·24-s + 2·25-s + 4·27-s − 5·32-s + 16·33-s − 34-s − 3·36-s + 8·38-s + 8·43-s − 8·44-s − 2·48-s + 2·49-s − 2·50-s + 2·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 9-s + 2.41·11-s − 0.577·12-s − 1/4·16-s + 0.242·17-s − 0.707·18-s − 1.83·19-s − 1.70·22-s + 1.22·24-s + 2/5·25-s + 0.769·27-s − 0.883·32-s + 2.78·33-s − 0.171·34-s − 1/2·36-s + 1.29·38-s + 1.21·43-s − 1.20·44-s − 0.288·48-s + 2/7·49-s − 0.282·50-s + 0.280·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.735936085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.735936085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.82580289195797930220355288431, −7.14257618232615439309453339322, −6.81089493407269657838307778384, −6.66687350936944381817930751537, −5.97184583023480145443864217661, −5.59334879276053026033607263499, −4.73157702271605766153507759357, −4.38066853020012071837140433024, −4.14871936049865192617863162664, −3.58805876328343905426425196162, −3.33506544840499141212096962726, −2.28251357059639089877751737038, −2.04655503075998176246419168666, −1.28950375172625955369776789729, −0.75526794503091583672922454443,
0.75526794503091583672922454443, 1.28950375172625955369776789729, 2.04655503075998176246419168666, 2.28251357059639089877751737038, 3.33506544840499141212096962726, 3.58805876328343905426425196162, 4.14871936049865192617863162664, 4.38066853020012071837140433024, 4.73157702271605766153507759357, 5.59334879276053026033607263499, 5.97184583023480145443864217661, 6.66687350936944381817930751537, 6.81089493407269657838307778384, 7.14257618232615439309453339322, 7.82580289195797930220355288431
