| L(s) = 1 | − 2·3-s − 5-s + 9-s + 11-s + 4·13-s + 2·15-s − 4·19-s + 6·23-s + 25-s + 4·27-s − 6·29-s + 8·31-s − 2·33-s + 2·37-s − 8·39-s − 6·41-s − 8·43-s − 45-s + 6·47-s − 6·53-s − 55-s + 8·57-s − 12·59-s − 2·61-s − 4·65-s + 10·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s − 0.824·53-s − 0.134·55-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.496·65-s + 1.22·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.179456983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179456983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.93043586430255, −13.98562859718106, −13.76473452421277, −12.88902830708611, −12.65309028744101, −12.02771024648962, −11.42775560722985, −11.10268357998618, −10.81669653425646, −10.10594118189696, −9.501304782506279, −8.719948466743006, −8.447058555565232, −7.762923299189426, −6.972108061422238, −6.480408070202109, −6.168606381052839, −5.423942361180331, −4.866974317088540, −4.345555512538527, −3.574622705222735, −3.049384527032988, −2.011144814773939, −1.154590564901268, −0.4827364686881440,
0.4827364686881440, 1.154590564901268, 2.011144814773939, 3.049384527032988, 3.574622705222735, 4.345555512538527, 4.866974317088540, 5.423942361180331, 6.168606381052839, 6.480408070202109, 6.972108061422238, 7.762923299189426, 8.447058555565232, 8.719948466743006, 9.501304782506279, 10.10594118189696, 10.81669653425646, 11.10268357998618, 11.42775560722985, 12.02771024648962, 12.65309028744101, 12.88902830708611, 13.76473452421277, 13.98562859718106, 14.93043586430255
