L(s) = 1 | − 2·11-s − 4·13-s − 17-s − 23-s + 2·29-s − 5·31-s + 4·37-s + 9·41-s + 4·43-s + 11·47-s + 6·59-s − 6·61-s − 3·71-s + 6·73-s − 15·79-s − 6·83-s − 9·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.603·11-s − 1.10·13-s − 0.242·17-s − 0.208·23-s + 0.371·29-s − 0.898·31-s + 0.657·37-s + 1.40·41-s + 0.609·43-s + 1.60·47-s + 0.781·59-s − 0.768·61-s − 0.356·71-s + 0.702·73-s − 1.68·79-s − 0.658·83-s − 0.953·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.428747706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428747706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−14.13394186068350, −13.25768077310560, −12.91760854112113, −12.40630717927540, −12.06021470710942, −11.29344719904160, −11.01259447102000, −10.33113676321037, −9.991401256531350, −9.359825870523408, −8.964057749458494, −8.342616950864122, −7.691759698725016, −7.359761993528545, −6.913350244328544, −6.039399237401304, −5.705319118983188, −5.097044372905382, −4.428762103346379, −4.083569251827442, −3.196639026428229, −2.527560148815436, −2.228297752472765, −1.229774995037621, −0.3952428620789228,
0.3952428620789228, 1.229774995037621, 2.228297752472765, 2.527560148815436, 3.196639026428229, 4.083569251827442, 4.428762103346379, 5.097044372905382, 5.705319118983188, 6.039399237401304, 6.913350244328544, 7.359761993528545, 7.691759698725016, 8.342616950864122, 8.964057749458494, 9.359825870523408, 9.991401256531350, 10.33113676321037, 11.01259447102000, 11.29344719904160, 12.06021470710942, 12.40630717927540, 12.91760854112113, 13.25768077310560, 14.13394186068350