| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 5·11-s − 4·13-s − 14-s + 16-s + 6·19-s + 20-s + 5·22-s − 4·25-s − 4·26-s − 28-s + 3·29-s + 5·31-s + 32-s − 35-s − 8·37-s + 6·38-s + 40-s − 6·41-s + 5·44-s + 2·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s + 1.06·22-s − 4/5·25-s − 0.784·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.169·35-s − 1.31·37-s + 0.973·38-s + 0.158·40-s − 0.937·41-s + 0.753·44-s + 0.291·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.527187611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.527187611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.76514205695436, −14.20063220106993, −13.94816107983911, −13.49518487337131, −12.80713807090413, −12.17747305313463, −11.75831143533978, −11.65081132378885, −10.62373863620523, −9.997079399581709, −9.746637662974222, −9.112035935855537, −8.521530259130941, −7.671966224160692, −7.165110791491630, −6.597114868211958, −6.201188872261140, −5.388116531650484, −5.013056294111485, −4.243526676374053, −3.603697302724432, −3.071894504723904, −2.257743055189562, −1.583694778418824, −0.7101466876417695,
0.7101466876417695, 1.583694778418824, 2.257743055189562, 3.071894504723904, 3.603697302724432, 4.243526676374053, 5.013056294111485, 5.388116531650484, 6.201188872261140, 6.597114868211958, 7.165110791491630, 7.671966224160692, 8.521530259130941, 9.112035935855537, 9.746637662974222, 9.997079399581709, 10.62373863620523, 11.65081132378885, 11.75831143533978, 12.17747305313463, 12.80713807090413, 13.49518487337131, 13.94816107983911, 14.20063220106993, 14.76514205695436
