| L(s) = 1 | − 2.82·3-s + 4.62·7-s − 19.0·9-s + 11·11-s − 85.1·13-s + 6.08·17-s + 52.0·19-s − 13.0·21-s + 183.·23-s + 130.·27-s + 140.·29-s − 250.·31-s − 31.0·33-s + 203.·37-s + 240.·39-s − 22.3·41-s + 117.·43-s + 275.·47-s − 321.·49-s − 17.2·51-s − 26.6·53-s − 147.·57-s + 515.·59-s − 693.·61-s − 87.9·63-s − 341.·67-s − 519.·69-s + ⋯ |
| L(s) = 1 | − 0.543·3-s + 0.249·7-s − 0.704·9-s + 0.301·11-s − 1.81·13-s + 0.0868·17-s + 0.628·19-s − 0.135·21-s + 1.66·23-s + 0.926·27-s + 0.899·29-s − 1.45·31-s − 0.164·33-s + 0.904·37-s + 0.987·39-s − 0.0850·41-s + 0.416·43-s + 0.855·47-s − 0.937·49-s − 0.0472·51-s − 0.0690·53-s − 0.342·57-s + 1.13·59-s − 1.45·61-s − 0.175·63-s − 0.622·67-s − 0.907·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 2.82T + 27T^{2} \) |
| 7 | \( 1 - 4.62T + 343T^{2} \) |
| 13 | \( 1 + 85.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.08T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 831.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 456.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 91.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 146.T + 9.12e5T^{2} \) |
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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
−8.304737327744647718538732077964, −7.38340723176083112933769185069, −6.87857255057538110907225005830, −5.80078198183209097151375104190, −5.14063529140469279279898214301, −4.52657354703493454863168756313, −3.18405185577162295602357450361, −2.43445860356636670105037446374, −1.06985008422778255595372539392, 0,
1.06985008422778255595372539392, 2.43445860356636670105037446374, 3.18405185577162295602357450361, 4.52657354703493454863168756313, 5.14063529140469279279898214301, 5.80078198183209097151375104190, 6.87857255057538110907225005830, 7.38340723176083112933769185069, 8.304737327744647718538732077964
