| L(s) = 1 | − 3-s + 9-s − 6·11-s + 13-s + 6·17-s + 5·19-s − 8·23-s − 27-s + 2·29-s − 4·31-s + 6·33-s + 37-s − 39-s − 6·41-s − 12·43-s − 12·47-s − 6·51-s + 6·53-s − 5·57-s + 8·59-s − 15·61-s − 5·67-s + 8·69-s − 6·71-s + 73-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 1.45·17-s + 1.14·19-s − 1.66·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 1.04·33-s + 0.164·37-s − 0.160·39-s − 0.937·41-s − 1.82·43-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.662·57-s + 1.04·59-s − 1.92·61-s − 0.610·67-s + 0.963·69-s − 0.712·71-s + 0.117·73-s − 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.29041741693713, −12.53181319168236, −12.24073709023576, −11.69346560676811, −11.45076986867105, −10.74570456356884, −10.18016876342700, −10.06279642858307, −9.727169143625372, −8.869033061908792, −8.264800011861084, −7.955460403417664, −7.485701908450039, −7.124407647657453, −6.262389622456099, −5.950537946568107, −5.385602648618705, −5.049597363675438, −4.628598726736305, −3.695673402834303, −3.310590889873801, −2.848960120575428, −1.939788011901327, −1.549157132262993, −0.6356746144320483, 0,
0.6356746144320483, 1.549157132262993, 1.939788011901327, 2.848960120575428, 3.310590889873801, 3.695673402834303, 4.628598726736305, 5.049597363675438, 5.385602648618705, 5.950537946568107, 6.262389622456099, 7.124407647657453, 7.485701908450039, 7.955460403417664, 8.264800011861084, 8.869033061908792, 9.727169143625372, 10.06279642858307, 10.18016876342700, 10.74570456356884, 11.45076986867105, 11.69346560676811, 12.24073709023576, 12.53181319168236, 13.29041741693713
