L(s) = 1 | + 3·5-s + 4·7-s + 13-s − 3·17-s − 4·19-s + 4·25-s − 9·29-s + 4·31-s + 12·35-s + 37-s + 6·41-s + 8·43-s + 12·47-s + 9·49-s + 6·53-s + 61-s + 3·65-s − 4·67-s + 12·71-s + 11·73-s + 16·79-s − 12·83-s − 9·85-s − 3·89-s + 4·91-s − 12·95-s + 2·97-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 1.51·7-s + 0.277·13-s − 0.727·17-s − 0.917·19-s + 4/5·25-s − 1.67·29-s + 0.718·31-s + 2.02·35-s + 0.164·37-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.128·61-s + 0.372·65-s − 0.488·67-s + 1.42·71-s + 1.28·73-s + 1.80·79-s − 1.31·83-s − 0.976·85-s − 0.317·89-s + 0.419·91-s − 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.190668903\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.190668903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.232374036135459680553193882425, −7.56796841047164741259196221788, −6.70609819233810081586523518704, −5.88525525346038032578810702567, −5.44593867080302146754053827061, −4.56777520805634839354449713902, −3.92139592663038120099322276304, −2.35182610767126161221653664716, −2.08634101847373408833873631943, −1.02356060707584348961647541129,
1.02356060707584348961647541129, 2.08634101847373408833873631943, 2.35182610767126161221653664716, 3.92139592663038120099322276304, 4.56777520805634839354449713902, 5.44593867080302146754053827061, 5.88525525346038032578810702567, 6.70609819233810081586523518704, 7.56796841047164741259196221788, 8.232374036135459680553193882425