| L(s) = 1 | + 3-s − 7-s − 2·9-s + 3·11-s − 3·13-s − 3·17-s + 2·19-s − 21-s + 23-s − 5·27-s + 3·29-s + 4·31-s + 3·33-s − 8·37-s − 3·39-s − 10·41-s + 4·43-s + 13·47-s + 49-s − 3·51-s + 2·57-s + 8·59-s − 10·61-s + 2·63-s − 6·67-s + 69-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.832·13-s − 0.727·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s − 1.31·37-s − 0.480·39-s − 1.56·41-s + 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.420·51-s + 0.264·57-s + 1.04·59-s − 1.28·61-s + 0.251·63-s − 0.733·67-s + 0.120·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.33761318671570, −13.98386370677391, −13.68937166852869, −13.05513478126767, −12.38291623045649, −11.95738000668031, −11.61232432109331, −10.97094501207935, −10.25092171893976, −9.951622505183890, −9.218573348926201, −8.800952496028867, −8.567195489366087, −7.764805612140842, −7.070521405047083, −6.908798122998405, −5.985157554170532, −5.695465357468699, −4.751785331372895, −4.416792669136602, −3.490591435033466, −3.175034208905933, −2.428728938774904, −1.884765994885854, −0.9053750014014550, 0,
0.9053750014014550, 1.884765994885854, 2.428728938774904, 3.175034208905933, 3.490591435033466, 4.416792669136602, 4.751785331372895, 5.695465357468699, 5.985157554170532, 6.908798122998405, 7.070521405047083, 7.764805612140842, 8.567195489366087, 8.800952496028867, 9.218573348926201, 9.951622505183890, 10.25092171893976, 10.97094501207935, 11.61232432109331, 11.95738000668031, 12.38291623045649, 13.05513478126767, 13.68937166852869, 13.98386370677391, 14.33761318671570
