| L(s) = 1 | − 4-s + 5·9-s − 12·11-s + 16-s − 4·19-s + 6·29-s + 16·31-s − 5·36-s + 18·41-s + 12·44-s + 12·59-s + 10·61-s − 64-s − 12·71-s + 4·76-s − 4·79-s + 16·81-s + 30·89-s − 60·99-s + 30·101-s − 22·109-s − 6·116-s + 86·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s − 3.61·11-s + 1/4·16-s − 0.917·19-s + 1.11·29-s + 2.87·31-s − 5/6·36-s + 2.81·41-s + 1.80·44-s + 1.56·59-s + 1.28·61-s − 1/8·64-s − 1.42·71-s + 0.458·76-s − 0.450·79-s + 16/9·81-s + 3.17·89-s − 6.03·99-s + 2.98·101-s − 2.10·109-s − 0.557·116-s + 7.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958078455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958078455\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.916073825779017114813652110429, −8.837296375002174507885969662412, −8.086935988207526195303649339415, −8.054545846937494541576996830331, −7.63705682903085284333294246140, −7.50801517098084994618186617547, −6.68867565488933977413309684184, −6.63693939152413596712306222011, −5.80466386298124727863835560445, −5.72403657437248080753190225713, −5.00305036389965495490252820250, −4.72476822876688739906650603138, −4.51649856121100240501898881631, −4.11225012261508123509731880090, −3.36130042430276758645330787870, −2.77646275272523582815325123908, −2.42005473408168570924055667029, −2.14919935820050753016894136363, −0.974535209220500395309237700336, −0.58471077136617434502516620810,
0.58471077136617434502516620810, 0.974535209220500395309237700336, 2.14919935820050753016894136363, 2.42005473408168570924055667029, 2.77646275272523582815325123908, 3.36130042430276758645330787870, 4.11225012261508123509731880090, 4.51649856121100240501898881631, 4.72476822876688739906650603138, 5.00305036389965495490252820250, 5.72403657437248080753190225713, 5.80466386298124727863835560445, 6.63693939152413596712306222011, 6.68867565488933977413309684184, 7.50801517098084994618186617547, 7.63705682903085284333294246140, 8.054545846937494541576996830331, 8.086935988207526195303649339415, 8.837296375002174507885969662412, 8.916073825779017114813652110429
