L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s + 3·12-s − 7·13-s + 16-s + 7·17-s − 6·18-s − 19-s + 8·23-s − 3·24-s − 5·25-s + 7·26-s + 9·27-s + 9·29-s + 2·31-s − 32-s − 7·34-s + 6·36-s + 3·37-s + 38-s − 21·39-s − 10·41-s + 10·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s + 0.866·12-s − 1.94·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s − 0.229·19-s + 1.66·23-s − 0.612·24-s − 25-s + 1.37·26-s + 1.73·27-s + 1.67·29-s + 0.359·31-s − 0.176·32-s − 1.20·34-s + 36-s + 0.493·37-s + 0.162·38-s − 3.36·39-s − 1.56·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.770941468\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.770941468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.343784787049288250151561526278, −7.59428648141978179934030198035, −7.41444755543721562166042908750, −6.50018362190960653186803724107, −5.25524637908827971867499931624, −4.48515576886248486520230602491, −3.34455255488757409394848077156, −2.81304263164298436582493772901, −2.10444152410754807673919512636, −0.971511438944262836171635213820,
0.971511438944262836171635213820, 2.10444152410754807673919512636, 2.81304263164298436582493772901, 3.34455255488757409394848077156, 4.48515576886248486520230602491, 5.25524637908827971867499931624, 6.50018362190960653186803724107, 7.41444755543721562166042908750, 7.59428648141978179934030198035, 8.343784787049288250151561526278