L(s) = 1 | + 7-s + 4·11-s + 13-s + 5·17-s − 2·19-s − 3·23-s − 5·25-s − 29-s + 9·31-s + 6·37-s − 2·41-s − 43-s + 2·47-s + 49-s + 9·53-s + 3·59-s − 2·61-s + 11·67-s − 5·71-s − 2·73-s + 4·77-s + 14·79-s − 12·83-s − 15·89-s + 91-s + 12·97-s − 6·101-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.277·13-s + 1.21·17-s − 0.458·19-s − 0.625·23-s − 25-s − 0.185·29-s + 1.61·31-s + 0.986·37-s − 0.312·41-s − 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.23·53-s + 0.390·59-s − 0.256·61-s + 1.34·67-s − 0.593·71-s − 0.234·73-s + 0.455·77-s + 1.57·79-s − 1.31·83-s − 1.58·89-s + 0.104·91-s + 1.21·97-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430774856\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430774856\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.147549622231165247983393077366, −7.42799562229350452640845684525, −6.57055765597438786848587784282, −6.00252378722893813603809380003, −5.28327987329702124669668451814, −4.22785079009045144006768014476, −3.84666015282243271147401636785, −2.77527515795993600777787809890, −1.74527430403008599847080379354, −0.868793283230493853483702913897,
0.868793283230493853483702913897, 1.74527430403008599847080379354, 2.77527515795993600777787809890, 3.84666015282243271147401636785, 4.22785079009045144006768014476, 5.28327987329702124669668451814, 6.00252378722893813603809380003, 6.57055765597438786848587784282, 7.42799562229350452640845684525, 8.147549622231165247983393077366