| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 6·11-s − 2·13-s + 15-s − 4·17-s + 2·19-s − 21-s + 23-s + 25-s − 27-s − 8·29-s + 6·31-s − 6·33-s − 35-s + 2·37-s + 2·39-s − 10·41-s − 2·43-s − 45-s − 2·47-s + 49-s + 4·51-s − 6·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.07·31-s − 1.04·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s − 1.56·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610302933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610302933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.51632771580910, −15.30176462277453, −14.59464493757268, −14.22681472808155, −13.45297507727217, −12.97661060123800, −12.14122568888010, −11.84796235939105, −11.33175363056361, −11.00542444363997, −10.05252923090778, −9.619937946120633, −8.937872708486090, −8.471508916682298, −7.620069507147592, −7.084341880999697, −6.518309312370282, −6.036987178012363, −5.019547133442101, −4.693402297455050, −3.880659004551816, −3.396933469939135, −2.202152818399062, −1.485118862528625, −0.5709222828007873,
0.5709222828007873, 1.485118862528625, 2.202152818399062, 3.396933469939135, 3.880659004551816, 4.693402297455050, 5.019547133442101, 6.036987178012363, 6.518309312370282, 7.084341880999697, 7.620069507147592, 8.471508916682298, 8.937872708486090, 9.619937946120633, 10.05252923090778, 11.00542444363997, 11.33175363056361, 11.84796235939105, 12.14122568888010, 12.97661060123800, 13.45297507727217, 14.22681472808155, 14.59464493757268, 15.30176462277453, 15.51632771580910
