| L(s) = 1 | + 3-s − 7-s + 9-s + 2·13-s + 6·17-s − 8·19-s − 21-s + 27-s − 6·29-s − 4·31-s − 10·37-s + 2·39-s − 6·41-s − 4·43-s + 49-s + 6·51-s − 6·53-s − 8·57-s + 12·59-s + 10·61-s − 63-s − 4·67-s + 12·71-s + 10·73-s + 8·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 0.218·21-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 1.64·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s − 0.125·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.232561212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232561212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.95785187191935, −14.56293093687707, −13.94121740365507, −13.44852828725361, −12.83833516108726, −12.55873418524206, −11.91537209503676, −11.24428450272660, −10.58635470883197, −10.26873622007054, −9.560345284440666, −9.117582368142585, −8.354136765551018, −8.184944846787444, −7.380024400428196, −6.724250106080020, −6.353631651853751, −5.409410938383966, −5.127804915829586, −3.916734554350276, −3.753503538885082, −3.079671759691536, −2.110657816294656, −1.664010788551633, −0.5282401016773331,
0.5282401016773331, 1.664010788551633, 2.110657816294656, 3.079671759691536, 3.753503538885082, 3.916734554350276, 5.127804915829586, 5.409410938383966, 6.353631651853751, 6.724250106080020, 7.380024400428196, 8.184944846787444, 8.354136765551018, 9.117582368142585, 9.560345284440666, 10.26873622007054, 10.58635470883197, 11.24428450272660, 11.91537209503676, 12.55873418524206, 12.83833516108726, 13.44852828725361, 13.94121740365507, 14.56293093687707, 14.95785187191935
