L(s) = 1 | − 2·11-s − 2·13-s − 2·17-s − 8·19-s + 2·23-s + 8·29-s − 8·31-s + 6·37-s + 6·41-s + 4·43-s + 4·47-s + 4·59-s + 2·61-s − 4·67-s + 6·71-s − 6·73-s − 16·83-s + 6·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.603·11-s − 0.554·13-s − 0.485·17-s − 1.83·19-s + 0.417·23-s + 1.48·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s + 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.712·71-s − 0.702·73-s − 1.75·83-s + 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.283839336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.283839336\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.03373723891968, −12.69335129775002, −12.47317095868627, −11.75282824144214, −11.12478176653102, −10.91823084350714, −10.33359198428350, −9.983878147420700, −9.317193044266665, −8.813394840857007, −8.502492558914800, −7.870449631236164, −7.388844919471091, −6.911809783380584, −6.355195924997873, −5.868332402117605, −5.350455559409488, −4.573369910752109, −4.416457077234503, −3.747143681770448, −2.934271103831510, −2.426659540040235, −2.065251066950893, −1.118681232199077, −0.3424696589389780,
0.3424696589389780, 1.118681232199077, 2.065251066950893, 2.426659540040235, 2.934271103831510, 3.747143681770448, 4.416457077234503, 4.573369910752109, 5.350455559409488, 5.868332402117605, 6.355195924997873, 6.911809783380584, 7.388844919471091, 7.870449631236164, 8.502492558914800, 8.813394840857007, 9.317193044266665, 9.983878147420700, 10.33359198428350, 10.91823084350714, 11.12478176653102, 11.75282824144214, 12.47317095868627, 12.69335129775002, 13.03373723891968