| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 11-s − 4·13-s − 15-s − 6·17-s − 2·21-s − 6·23-s + 25-s + 27-s − 10·29-s − 8·31-s + 33-s + 2·35-s − 8·37-s − 4·39-s + 10·41-s − 6·43-s − 45-s − 6·47-s − 3·49-s − 6·51-s + 4·53-s − 55-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.174·33-s + 0.338·35-s − 1.31·37-s − 0.640·39-s + 1.56·41-s − 0.914·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.549·53-s − 0.134·55-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.14627328872661, −12.73982405559493, −12.31724169037755, −11.70727097854310, −11.31224368035631, −10.89169960975712, −10.15648794139347, −9.859445997481200, −9.348740681081778, −8.983528195914429, −8.510072161495225, −7.911585036104937, −7.466347441719183, −6.871763906567733, −6.784703397513054, −5.959608648461992, −5.418728146902078, −4.894729444359682, −4.181488897952215, −3.780215982850719, −3.461027981784682, −2.652625914298840, −2.010950150596004, −1.863370685026526, −0.5440115928556575, 0,
0.5440115928556575, 1.863370685026526, 2.010950150596004, 2.652625914298840, 3.461027981784682, 3.780215982850719, 4.181488897952215, 4.894729444359682, 5.418728146902078, 5.959608648461992, 6.784703397513054, 6.871763906567733, 7.466347441719183, 7.911585036104937, 8.510072161495225, 8.983528195914429, 9.348740681081778, 9.859445997481200, 10.15648794139347, 10.89169960975712, 11.31224368035631, 11.70727097854310, 12.31724169037755, 12.73982405559493, 13.14627328872661
