| L(s) = 1 | + 3-s − 2·4-s − 5-s + 2·7-s + 9-s − 6·11-s − 2·12-s + 2·13-s − 15-s + 4·16-s − 3·17-s − 19-s + 2·20-s + 2·21-s − 9·23-s + 25-s + 27-s − 4·28-s + 3·29-s − 4·31-s − 6·33-s − 2·35-s − 2·36-s − 7·37-s + 2·39-s + 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.229·19-s + 0.447·20-s + 0.436·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.557·29-s − 0.718·31-s − 1.04·33-s − 0.338·35-s − 1/3·36-s − 1.15·37-s + 0.320·39-s + 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.469806900331237576847704433820, −8.387459012074038213015262648375, −8.214779397591055351124054522191, −7.40618624472063606172206788204, −5.95060752796028186161644735643, −4.95941691053001603518430597137, −4.28487919756175465855132671735, −3.24494354251730337838642008272, −1.92409682240405377626184831465, 0,
1.92409682240405377626184831465, 3.24494354251730337838642008272, 4.28487919756175465855132671735, 4.95941691053001603518430597137, 5.95060752796028186161644735643, 7.40618624472063606172206788204, 8.214779397591055351124054522191, 8.387459012074038213015262648375, 9.469806900331237576847704433820
