| L(s) = 1 | + 3-s + 9-s − 2·11-s + 6·13-s + 8·19-s − 3·23-s + 27-s − 4·29-s − 10·31-s − 2·33-s − 5·37-s + 6·39-s − 9·41-s − 6·43-s + 4·47-s − 7·49-s + 3·53-s + 8·57-s + 3·59-s + 13·61-s + 2·67-s − 3·69-s − 5·71-s − 4·79-s + 81-s + 9·83-s − 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 1.83·19-s − 0.625·23-s + 0.192·27-s − 0.742·29-s − 1.79·31-s − 0.348·33-s − 0.821·37-s + 0.960·39-s − 1.40·41-s − 0.914·43-s + 0.583·47-s − 49-s + 0.412·53-s + 1.05·57-s + 0.390·59-s + 1.66·61-s + 0.244·67-s − 0.361·69-s − 0.593·71-s − 0.450·79-s + 1/9·81-s + 0.987·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850806294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850806294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.20070264726014, −13.01726560809182, −12.23777799714724, −11.72201463706526, −11.33791602511055, −10.86933537741072, −10.17499178986361, −10.03024443183075, −9.189241722091931, −8.989426175605756, −8.403834326083528, −7.878397976027050, −7.571659263861334, −6.869631458311833, −6.529123755569103, −5.638721948930192, −5.411364962390524, −4.933526144333586, −3.846965290438450, −3.711955392532476, −3.247397504517920, −2.524915454128996, −1.707621782915374, −1.424731892438474, −0.4603939863180981,
0.4603939863180981, 1.424731892438474, 1.707621782915374, 2.524915454128996, 3.247397504517920, 3.711955392532476, 3.846965290438450, 4.933526144333586, 5.411364962390524, 5.638721948930192, 6.529123755569103, 6.869631458311833, 7.571659263861334, 7.878397976027050, 8.403834326083528, 8.989426175605756, 9.189241722091931, 10.03024443183075, 10.17499178986361, 10.86933537741072, 11.33791602511055, 11.72201463706526, 12.23777799714724, 13.01726560809182, 13.20070264726014
