| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·11-s + 2·12-s + 16-s + 8·23-s + 4·27-s − 16·31-s + 4·33-s + 3·36-s − 4·37-s + 2·44-s + 16·47-s + 2·48-s − 10·49-s − 12·53-s + 20·59-s + 64-s + 16·67-s + 16·69-s + 24·71-s + 5·81-s − 20·89-s + 8·92-s − 32·93-s + 16·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 0.603·11-s + 0.577·12-s + 1/4·16-s + 1.66·23-s + 0.769·27-s − 2.87·31-s + 0.696·33-s + 1/2·36-s − 0.657·37-s + 0.301·44-s + 2.33·47-s + 0.288·48-s − 1.42·49-s − 1.64·53-s + 2.60·59-s + 1/8·64-s + 1.95·67-s + 1.92·69-s + 2.84·71-s + 5/9·81-s − 2.11·89-s + 0.834·92-s − 3.31·93-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.748786396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.748786396\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.47947512213883760565427163251, −7.25165118480744247888197632653, −6.82388856195908314993907699373, −6.64959962043719359186062300669, −5.93970823062035349080632113655, −5.38347445400098472140318732151, −5.13544146251726925083565635926, −4.54438284729604921144723239547, −3.79143046769365773246334925917, −3.62073104001028637782990513939, −3.26722466254523526853639387287, −2.45815773509881651106722995929, −2.15599683563985558220964117330, −1.54428972648825485047162953199, −0.797857127952447136411610633543,
0.797857127952447136411610633543, 1.54428972648825485047162953199, 2.15599683563985558220964117330, 2.45815773509881651106722995929, 3.26722466254523526853639387287, 3.62073104001028637782990513939, 3.79143046769365773246334925917, 4.54438284729604921144723239547, 5.13544146251726925083565635926, 5.38347445400098472140318732151, 5.93970823062035349080632113655, 6.64959962043719359186062300669, 6.82388856195908314993907699373, 7.25165118480744247888197632653, 7.47947512213883760565427163251
