| L(s) = 1 | − 4·7-s + 4·11-s − 13-s + 4·17-s − 7·19-s − 4·23-s + 5·29-s + 4·31-s − 9·37-s + 5·41-s + 10·43-s − 3·47-s + 9·49-s + 9·53-s − 6·59-s − 4·61-s + 7·67-s + 15·71-s + 12·73-s − 16·77-s + 7·79-s + 6·83-s − 14·89-s + 4·91-s − 16·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s − 1.60·19-s − 0.834·23-s + 0.928·29-s + 0.718·31-s − 1.47·37-s + 0.780·41-s + 1.52·43-s − 0.437·47-s + 9/7·49-s + 1.23·53-s − 0.781·59-s − 0.512·61-s + 0.855·67-s + 1.78·71-s + 1.40·73-s − 1.82·77-s + 0.787·79-s + 0.658·83-s − 1.48·89-s + 0.419·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976774442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976774442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.02449820123573, −12.41922816136225, −12.28923646687510, −12.06615835473791, −11.10480912537983, −10.80583429891926, −10.09742414875132, −9.901386576602594, −9.351029504183199, −8.979182251051621, −8.312304858051716, −8.007961563203848, −7.134142714360809, −6.767007412252319, −6.399004577034397, −5.941297626815088, −5.456296789027598, −4.573499913442953, −4.144905685961076, −3.621440817957012, −3.191344318619887, −2.440942636693795, −1.971915061897505, −1.003799884822183, −0.4638439214285162,
0.4638439214285162, 1.003799884822183, 1.971915061897505, 2.440942636693795, 3.191344318619887, 3.621440817957012, 4.144905685961076, 4.573499913442953, 5.456296789027598, 5.941297626815088, 6.399004577034397, 6.767007412252319, 7.134142714360809, 8.007961563203848, 8.312304858051716, 8.979182251051621, 9.351029504183199, 9.901386576602594, 10.09742414875132, 10.80583429891926, 11.10480912537983, 12.06615835473791, 12.28923646687510, 12.41922816136225, 13.02449820123573
