| L(s) = 1 | − 2-s − 0.445·3-s + 4-s + 0.445·6-s − 0.246·7-s − 8-s − 2.80·9-s + 0.801·11-s − 0.445·12-s + 0.246·14-s + 16-s + 1.55·17-s + 2.80·18-s + 1.55·19-s + 0.109·21-s − 0.801·22-s + 0.692·23-s + 0.445·24-s + 2.58·27-s − 0.246·28-s − 0.286·29-s − 4.15·31-s − 32-s − 0.356·33-s − 1.55·34-s − 2.80·36-s + 8.28·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.256·3-s + 0.5·4-s + 0.181·6-s − 0.0933·7-s − 0.353·8-s − 0.933·9-s + 0.241·11-s − 0.128·12-s + 0.0660·14-s + 0.250·16-s + 0.377·17-s + 0.660·18-s + 0.356·19-s + 0.0239·21-s − 0.170·22-s + 0.144·23-s + 0.0908·24-s + 0.496·27-s − 0.0466·28-s − 0.0531·29-s − 0.746·31-s − 0.176·32-s − 0.0621·33-s − 0.266·34-s − 0.466·36-s + 1.36·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.445T + 3T^{2} \) |
| 7 | \( 1 + 0.246T + 7T^{2} \) |
| 11 | \( 1 - 0.801T + 11T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 0.692T + 23T^{2} \) |
| 29 | \( 1 + 0.286T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 - 8.28T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 - 0.868T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 - 4.72T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−7.59089784007517731999798278268, −6.74730050242341074552812407516, −6.18337163973340963566248238733, −5.48217844215468011035491972100, −4.80873641882095892216649790740, −3.66087032845617400684103400013, −3.02655899927424967141701460542, −2.11047990473674303354062127391, −1.07813277628567484538340051660, 0,
1.07813277628567484538340051660, 2.11047990473674303354062127391, 3.02655899927424967141701460542, 3.66087032845617400684103400013, 4.80873641882095892216649790740, 5.48217844215468011035491972100, 6.18337163973340963566248238733, 6.74730050242341074552812407516, 7.59089784007517731999798278268
