| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 7-s + 8·10-s − 6·11-s + 13-s + 2·14-s − 4·16-s + 17-s − 7·19-s + 8·20-s − 12·22-s + 4·23-s + 11·25-s + 2·26-s + 2·28-s + 6·29-s − 8·31-s − 8·32-s + 2·34-s + 4·35-s + 37-s − 14·38-s + 4·43-s − 12·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 0.377·7-s + 2.52·10-s − 1.80·11-s + 0.277·13-s + 0.534·14-s − 16-s + 0.242·17-s − 1.60·19-s + 1.78·20-s − 2.55·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s + 0.676·35-s + 0.164·37-s − 2.27·38-s + 0.609·43-s − 1.80·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364460632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364460632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−10.90508481410394760093183518479, −10.51882698778784990220643195703, −9.355188181832179362346103161628, −8.379898543961470967191694971495, −6.92940427570655812640731061069, −5.93408556270929922136180838803, −5.37713357577775960919289343412, −4.56441091173163015164546337868, −2.90936379528655043489992111903, −2.09364906131822284439528208110,
2.09364906131822284439528208110, 2.90936379528655043489992111903, 4.56441091173163015164546337868, 5.37713357577775960919289343412, 5.93408556270929922136180838803, 6.92940427570655812640731061069, 8.379898543961470967191694971495, 9.355188181832179362346103161628, 10.51882698778784990220643195703, 10.90508481410394760093183518479
