| L(s) = 1 | − 2·11-s − 2·13-s + 4·17-s − 19-s − 2·23-s − 5·25-s + 6·29-s − 4·31-s + 2·37-s − 2·41-s − 4·43-s − 10·47-s − 7·49-s + 6·53-s − 4·59-s + 2·61-s − 4·67-s − 4·71-s − 6·73-s − 8·79-s + 2·83-s + 6·89-s − 10·97-s + 4·101-s − 12·107-s − 2·109-s + 6·113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.229·19-s − 0.417·23-s − 25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.45·47-s − 49-s + 0.824·53-s − 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.474·71-s − 0.702·73-s − 0.900·79-s + 0.219·83-s + 0.635·89-s − 1.01·97-s + 0.398·101-s − 1.16·107-s − 0.191·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.255845507164553215307600810894, −7.82242986665710365233514054067, −6.97720773946475230429853816181, −6.09673641307441141291598153625, −5.33173854282618268823399412104, −4.56969481089501650056919417803, −3.54293501179707213566770305911, −2.66331292361056007770031824402, −1.56702608619439033692515095121, 0,
1.56702608619439033692515095121, 2.66331292361056007770031824402, 3.54293501179707213566770305911, 4.56969481089501650056919417803, 5.33173854282618268823399412104, 6.09673641307441141291598153625, 6.97720773946475230429853816181, 7.82242986665710365233514054067, 8.255845507164553215307600810894
