| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s − 6·6-s + 6·9-s + 11-s − 6·12-s − 3·13-s − 4·16-s + 3·17-s + 12·18-s + 6·19-s + 2·22-s + 4·23-s − 6·26-s − 9·27-s − 29-s + 6·31-s − 8·32-s − 3·33-s + 6·34-s + 12·36-s + 12·38-s + 9·39-s + 6·41-s + 6·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 4-s − 2.44·6-s + 2·9-s + 0.301·11-s − 1.73·12-s − 0.832·13-s − 16-s + 0.727·17-s + 2.82·18-s + 1.37·19-s + 0.426·22-s + 0.834·23-s − 1.17·26-s − 1.73·27-s − 0.185·29-s + 1.07·31-s − 1.41·32-s − 0.522·33-s + 1.02·34-s + 2·36-s + 1.94·38-s + 1.44·39-s + 0.937·41-s + 0.914·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.900832286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900832286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.997701708121623035643371382330, −9.172428119455054892701886762610, −7.51763973241378139376889112452, −6.91586626245215922674411076526, −5.97082893018481121134513026433, −5.44805135703077001763611976775, −4.80361288782423271836550841071, −3.98676991130909907422178128989, −2.75417149519990387874745921492, −0.932413867239647883867483465341,
0.932413867239647883867483465341, 2.75417149519990387874745921492, 3.98676991130909907422178128989, 4.80361288782423271836550841071, 5.44805135703077001763611976775, 5.97082893018481121134513026433, 6.91586626245215922674411076526, 7.51763973241378139376889112452, 9.172428119455054892701886762610, 9.997701708121623035643371382330
