| L(s) = 1 | + 3·3-s − 6·5-s + 20·7-s + 9·9-s − 24·11-s − 13·13-s − 18·15-s − 30·17-s + 16·19-s + 60·21-s − 72·23-s − 89·25-s + 27·27-s + 282·29-s + 164·31-s − 72·33-s − 120·35-s − 110·37-s − 39·39-s − 126·41-s − 164·43-s − 54·45-s − 204·47-s + 57·49-s − 90·51-s + 738·53-s + 144·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.536·5-s + 1.07·7-s + 1/3·9-s − 0.657·11-s − 0.277·13-s − 0.309·15-s − 0.428·17-s + 0.193·19-s + 0.623·21-s − 0.652·23-s − 0.711·25-s + 0.192·27-s + 1.80·29-s + 0.950·31-s − 0.379·33-s − 0.579·35-s − 0.488·37-s − 0.160·39-s − 0.479·41-s − 0.581·43-s − 0.178·45-s − 0.633·47-s + 0.166·49-s − 0.247·51-s + 1.91·53-s + 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 282 T + p^{3} T^{2} \) |
| 31 | \( 1 - 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 + 120 T + p^{3} T^{2} \) |
| 61 | \( 1 + 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 848 T + p^{3} T^{2} \) |
| 71 | \( 1 - 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1096 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
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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
−8.332000882195298676902530724457, −7.60183396981261854466321704808, −6.86963171070988769960099753345, −5.79212091346365461341625292036, −4.77004707883429040554838275059, −4.33308202003223622669385089414, −3.19130007233440145044251208007, −2.32823925710184006128101027239, −1.34199051385831543847779805331, 0,
1.34199051385831543847779805331, 2.32823925710184006128101027239, 3.19130007233440145044251208007, 4.33308202003223622669385089414, 4.77004707883429040554838275059, 5.79212091346365461341625292036, 6.86963171070988769960099753345, 7.60183396981261854466321704808, 8.332000882195298676902530724457
