| L(s) = 1 | + 3-s + 2·4-s − 5-s + 3·9-s + 3·11-s + 2·12-s − 10·13-s − 15-s + 3·17-s + 2·19-s − 2·20-s + 6·23-s + 8·27-s + 6·29-s − 4·31-s + 3·33-s + 6·36-s − 2·37-s − 10·39-s + 24·41-s − 20·43-s + 6·44-s − 3·45-s + 9·47-s + 3·51-s − 20·52-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s − 0.447·5-s + 9-s + 0.904·11-s + 0.577·12-s − 2.77·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1.53·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s + 36-s − 0.328·37-s − 1.60·39-s + 3.74·41-s − 3.04·43-s + 0.904·44-s − 0.447·45-s + 1.31·47-s + 0.420·51-s − 2.77·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.083808682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.083808682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−12.32392450593324665056395344160, −11.88970189943156110422580026500, −11.46080155353028838978960464645, −11.00288659065086848832981884739, −10.28556129907529394333505686566, −9.952359872216849956089022052653, −9.484976221237438958588851238989, −9.084317711588151097027953548515, −8.352192050458797959516434547587, −7.70998033654721587309174344678, −7.24233835928023693117242248699, −7.07274899484813142048842616642, −6.61123773043983510273189327447, −5.71690850778224111057713378588, −4.74644062665126137294936604230, −4.69368961275903088970943567662, −3.67353999802657571057822330118, −2.81610322323524882372353158831, −2.46794845066265842749121387058, −1.30773830699806101421822577632,
1.30773830699806101421822577632, 2.46794845066265842749121387058, 2.81610322323524882372353158831, 3.67353999802657571057822330118, 4.69368961275903088970943567662, 4.74644062665126137294936604230, 5.71690850778224111057713378588, 6.61123773043983510273189327447, 7.07274899484813142048842616642, 7.24233835928023693117242248699, 7.70998033654721587309174344678, 8.352192050458797959516434547587, 9.084317711588151097027953548515, 9.484976221237438958588851238989, 9.952359872216849956089022052653, 10.28556129907529394333505686566, 11.00288659065086848832981884739, 11.46080155353028838978960464645, 11.88970189943156110422580026500, 12.32392450593324665056395344160
