L(s) = 1 | + 5-s − 4·7-s − 11-s − 5·13-s + 3·17-s + 3·23-s + 25-s − 8·29-s + 31-s − 4·35-s − 8·37-s + 41-s − 43-s + 9·49-s + 3·53-s − 55-s − 8·61-s − 5·65-s − 9·67-s + 4·71-s + 4·73-s + 4·77-s − 8·79-s − 15·83-s + 3·85-s + 16·89-s + 20·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s − 1.38·13-s + 0.727·17-s + 0.625·23-s + 1/5·25-s − 1.48·29-s + 0.179·31-s − 0.676·35-s − 1.31·37-s + 0.156·41-s − 0.152·43-s + 9/7·49-s + 0.412·53-s − 0.134·55-s − 1.02·61-s − 0.620·65-s − 1.09·67-s + 0.474·71-s + 0.468·73-s + 0.455·77-s − 0.900·79-s − 1.64·83-s + 0.325·85-s + 1.69·89-s + 2.09·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3365377510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3365377510\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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
−13.46424267538208, −12.99663377593548, −12.62971151880650, −12.20162025468332, −11.77837890164362, −11.02188530305385, −10.43596397199069, −10.14272369871366, −9.578357218643628, −9.317248348119743, −8.836746512779061, −8.038240257269947, −7.489356669705751, −7.029838609594209, −6.670411920348798, −5.962758356368347, −5.471160154497414, −5.128085980397649, −4.328628698390012, −3.686166461815185, −3.056822305049807, −2.733237684674545, −2.002205967494154, −1.238666593362133, −0.1754692497967022,
0.1754692497967022, 1.238666593362133, 2.002205967494154, 2.733237684674545, 3.056822305049807, 3.686166461815185, 4.328628698390012, 5.128085980397649, 5.471160154497414, 5.962758356368347, 6.670411920348798, 7.029838609594209, 7.489356669705751, 8.038240257269947, 8.836746512779061, 9.317248348119743, 9.578357218643628, 10.14272369871366, 10.43596397199069, 11.02188530305385, 11.77837890164362, 12.20162025468332, 12.62971151880650, 12.99663377593548, 13.46424267538208