| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·11-s − 6·13-s + 14-s + 16-s + 4·17-s − 6·19-s − 20-s − 2·22-s − 23-s + 25-s + 6·26-s − 28-s − 6·31-s − 32-s − 4·34-s + 35-s + 6·37-s + 6·38-s + 40-s + 6·41-s + 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.986·37-s + 0.973·38-s + 0.158·40-s + 0.937·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6853571538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6853571538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.28603724090797, −15.64548445103867, −14.81437322324813, −14.68815451258266, −14.15746597007291, −13.00107891036123, −12.67599839079954, −12.12242674195290, −11.58780483760778, −10.95654755385178, −10.25740305736604, −9.845780443445297, −9.190580340849156, −8.732479740146389, −7.866074045732341, −7.447132078457699, −6.943441069203531, −6.116249920039173, −5.591519933648219, −4.541200316236048, −4.065490814495166, −3.071876164300933, −2.450607063286054, −1.535036111796863, −0.4039972896074559,
0.4039972896074559, 1.535036111796863, 2.450607063286054, 3.071876164300933, 4.065490814495166, 4.541200316236048, 5.591519933648219, 6.116249920039173, 6.943441069203531, 7.447132078457699, 7.866074045732341, 8.732479740146389, 9.190580340849156, 9.845780443445297, 10.25740305736604, 10.95654755385178, 11.58780483760778, 12.12242674195290, 12.67599839079954, 13.00107891036123, 14.15746597007291, 14.68815451258266, 14.81437322324813, 15.64548445103867, 16.28603724090797
