| L(s) = 1 | + 2·3-s − 3·9-s − 14·27-s + 8·31-s − 8·37-s + 6·41-s − 16·43-s − 2·49-s − 24·53-s + 18·67-s − 32·71-s − 8·79-s − 4·81-s − 2·83-s − 26·89-s + 16·93-s + 26·107-s − 16·111-s + 13·121-s + 12·123-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s − 2.69·27-s + 1.43·31-s − 1.31·37-s + 0.937·41-s − 2.43·43-s − 2/7·49-s − 3.29·53-s + 2.19·67-s − 3.79·71-s − 0.900·79-s − 4/9·81-s − 0.219·83-s − 2.75·89-s + 1.65·93-s + 2.51·107-s − 1.51·111-s + 1.18·121-s + 1.08·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.329·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230773999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230773999\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 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
−8.894963425889064803651097854305, −8.468323593157476187460836922501, −8.201719620958858160263906062570, −7.912797010012268664280487930204, −7.48904166211153461211905980389, −7.06131983411534643152493473078, −6.60678698787886229234535438840, −6.15036726561459343238310876851, −5.97761365514370637849146937989, −5.28842577470225479057146565050, −5.16825691585519940113257088253, −4.37948113771519708562207378527, −4.32428036316403344048825060858, −3.32670329954205519514977744063, −3.30373326282026763388137525611, −2.96276888779833845617795639777, −2.44537855772120569958870141478, −1.81464293586862136607624154828, −1.45588449903174021487327293218, −0.29715567679261336141009170021,
0.29715567679261336141009170021, 1.45588449903174021487327293218, 1.81464293586862136607624154828, 2.44537855772120569958870141478, 2.96276888779833845617795639777, 3.30373326282026763388137525611, 3.32670329954205519514977744063, 4.32428036316403344048825060858, 4.37948113771519708562207378527, 5.16825691585519940113257088253, 5.28842577470225479057146565050, 5.97761365514370637849146937989, 6.15036726561459343238310876851, 6.60678698787886229234535438840, 7.06131983411534643152493473078, 7.48904166211153461211905980389, 7.912797010012268664280487930204, 8.201719620958858160263906062570, 8.468323593157476187460836922501, 8.894963425889064803651097854305
