| L(s) = 1 | − 3-s − 2·5-s + 7-s + 3·9-s − 3·11-s + 2·13-s + 2·15-s + 3·17-s − 5·19-s − 21-s − 9·23-s + 3·25-s − 8·27-s + 9·29-s + 16·31-s + 3·33-s − 2·35-s + 7·37-s − 2·39-s − 3·41-s + 43-s − 6·45-s + 7·49-s − 3·51-s + 12·53-s + 6·55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 9-s − 0.904·11-s + 0.554·13-s + 0.516·15-s + 0.727·17-s − 1.14·19-s − 0.218·21-s − 1.87·23-s + 3/5·25-s − 1.53·27-s + 1.67·29-s + 2.87·31-s + 0.522·33-s − 0.338·35-s + 1.15·37-s − 0.320·39-s − 0.468·41-s + 0.152·43-s − 0.894·45-s + 49-s − 0.420·51-s + 1.64·53-s + 0.809·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005156350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005156350\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−12.00492380768242560403405660627, −11.98305682037253463131741967335, −11.35753933106026292011482398084, −10.73252545284597420117471941249, −10.30509448922771641790855168684, −10.12987672505612879469113749589, −9.545910646661125405353337508357, −8.547322484348089346348842264120, −8.282733700546634353088809613062, −7.907038550521635541879595245296, −7.42147685506373683673133857360, −6.72667613616255717497304742582, −6.12218659300924945656328048533, −5.80750831831955318944786424904, −4.83472529881436202340352022996, −4.37497526960047177461463069333, −4.05170959812265054789947377536, −3.03568569313595587159094219180, −2.15027365314411262605981248312, −0.856557334826550739761263451070,
0.856557334826550739761263451070, 2.15027365314411262605981248312, 3.03568569313595587159094219180, 4.05170959812265054789947377536, 4.37497526960047177461463069333, 4.83472529881436202340352022996, 5.80750831831955318944786424904, 6.12218659300924945656328048533, 6.72667613616255717497304742582, 7.42147685506373683673133857360, 7.907038550521635541879595245296, 8.282733700546634353088809613062, 8.547322484348089346348842264120, 9.545910646661125405353337508357, 10.12987672505612879469113749589, 10.30509448922771641790855168684, 10.73252545284597420117471941249, 11.35753933106026292011482398084, 11.98305682037253463131741967335, 12.00492380768242560403405660627
