L(s) = 1 | + 4·5-s + 3·13-s − 4·17-s − 7·19-s − 4·23-s + 11·25-s + 8·29-s + 5·31-s + 3·37-s + 8·41-s + 11·43-s + 4·47-s − 4·53-s + 12·59-s + 2·61-s + 12·65-s − 3·67-s − 12·71-s − 73-s + 79-s + 12·83-s − 16·85-s + 8·89-s − 28·95-s + 2·97-s − 3·103-s + 12·107-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.832·13-s − 0.970·17-s − 1.60·19-s − 0.834·23-s + 11/5·25-s + 1.48·29-s + 0.898·31-s + 0.493·37-s + 1.24·41-s + 1.67·43-s + 0.583·47-s − 0.549·53-s + 1.56·59-s + 0.256·61-s + 1.48·65-s − 0.366·67-s − 1.42·71-s − 0.117·73-s + 0.112·79-s + 1.31·83-s − 1.73·85-s + 0.847·89-s − 2.87·95-s + 0.203·97-s − 0.295·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.767220697\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.767220697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−8.803724209978870762479299082629, −7.978349303612857129811437752380, −6.76093763245926251492555053021, −6.19598722196138351939834316424, −5.88488609878110173493890067360, −4.74110169212235698613072750022, −4.08343580165689208704659809220, −2.61409889321325522445104600531, −2.17510792353690678574052363640, −1.02309438809947068717472620356,
1.02309438809947068717472620356, 2.17510792353690678574052363640, 2.61409889321325522445104600531, 4.08343580165689208704659809220, 4.74110169212235698613072750022, 5.88488609878110173493890067360, 6.19598722196138351939834316424, 6.76093763245926251492555053021, 7.978349303612857129811437752380, 8.803724209978870762479299082629