| L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s + 2·13-s − 2·15-s − 6·17-s − 25-s + 27-s + 2·29-s + 33-s + 10·37-s + 2·39-s + 6·41-s − 8·43-s − 2·45-s + 4·47-s − 7·49-s − 6·51-s + 6·53-s − 2·55-s + 12·59-s − 2·61-s − 4·65-s + 4·67-s − 12·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s + 0.583·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.130173564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130173564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.95876516756564, −12.28944076523089, −11.68608200471137, −11.36698067087391, −11.15435391265521, −10.36179737927967, −10.06044909151350, −9.367419789687481, −9.020360770099819, −8.451595257147999, −8.264275028434090, −7.622398617787910, −7.204095931710625, −6.728584224966410, −6.170278012210577, −5.764057017197241, −4.878504994608837, −4.459220038236394, −4.025212465785587, −3.640911956944770, −2.938140374439337, −2.456569855706677, −1.833433731196428, −1.104031871349677, −0.4055201172426983,
0.4055201172426983, 1.104031871349677, 1.833433731196428, 2.456569855706677, 2.938140374439337, 3.640911956944770, 4.025212465785587, 4.459220038236394, 4.878504994608837, 5.764057017197241, 6.170278012210577, 6.728584224966410, 7.204095931710625, 7.622398617787910, 8.264275028434090, 8.451595257147999, 9.020360770099819, 9.367419789687481, 10.06044909151350, 10.36179737927967, 11.15435391265521, 11.36698067087391, 11.68608200471137, 12.28944076523089, 12.95876516756564
