L(s) = 1 | − 2·4-s + 7-s + 11-s + 13-s + 4·16-s − 6·17-s − 7·19-s + 6·23-s − 2·28-s + 6·29-s − 7·31-s − 2·37-s + 6·41-s + 43-s − 2·44-s − 6·49-s − 2·52-s − 6·53-s + 5·61-s − 8·64-s − 5·67-s + 12·68-s + 12·71-s − 14·73-s + 14·76-s + 77-s − 4·79-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s + 0.301·11-s + 0.277·13-s + 16-s − 1.45·17-s − 1.60·19-s + 1.25·23-s − 0.377·28-s + 1.11·29-s − 1.25·31-s − 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.301·44-s − 6/7·49-s − 0.277·52-s − 0.824·53-s + 0.640·61-s − 64-s − 0.610·67-s + 1.45·68-s + 1.42·71-s − 1.63·73-s + 1.60·76-s + 0.113·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.768041607540139167824161107174, −8.001893150916943033064179219920, −6.94177064779494456719577763043, −6.25997416322047035967534085706, −5.23058676244714842631111750230, −4.48881609881667067649164980343, −3.93260401970958534385260320255, −2.69203750343984022101198568163, −1.44625488225869633667383305184, 0,
1.44625488225869633667383305184, 2.69203750343984022101198568163, 3.93260401970958534385260320255, 4.48881609881667067649164980343, 5.23058676244714842631111750230, 6.25997416322047035967534085706, 6.94177064779494456719577763043, 8.001893150916943033064179219920, 8.768041607540139167824161107174