| L(s) = 1 | + 2-s − 7·4-s + 5·5-s + 7·7-s − 15·8-s + 5·10-s + 11-s + 13·13-s + 7·14-s + 41·16-s − 19·17-s − 117·19-s − 35·20-s + 22-s + 141·23-s − 100·25-s + 13·26-s − 49·28-s + 131·29-s − 128·31-s + 161·32-s − 19·34-s + 35·35-s + 55·37-s − 117·38-s − 75·40-s − 201·43-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s + 0.447·5-s + 0.377·7-s − 0.662·8-s + 0.158·10-s + 0.0274·11-s + 0.277·13-s + 0.133·14-s + 0.640·16-s − 0.271·17-s − 1.41·19-s − 0.391·20-s + 0.00969·22-s + 1.27·23-s − 4/5·25-s + 0.0980·26-s − 0.330·28-s + 0.838·29-s − 0.741·31-s + 0.889·32-s − 0.0958·34-s + 0.169·35-s + 0.244·37-s − 0.499·38-s − 0.296·40-s − 0.712·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - T + p^{3} T^{2} \) |
| 17 | \( 1 + 19 T + p^{3} T^{2} \) |
| 19 | \( 1 + 117 T + p^{3} T^{2} \) |
| 23 | \( 1 - 141 T + p^{3} T^{2} \) |
| 29 | \( 1 - 131 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 55 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 201 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 510 T + p^{3} T^{2} \) |
| 59 | \( 1 - 156 T + p^{3} T^{2} \) |
| 61 | \( 1 + 845 T + p^{3} T^{2} \) |
| 67 | \( 1 + 470 T + p^{3} T^{2} \) |
| 71 | \( 1 + 324 T + p^{3} T^{2} \) |
| 73 | \( 1 + 373 T + p^{3} T^{2} \) |
| 79 | \( 1 + 526 T + p^{3} T^{2} \) |
| 83 | \( 1 + 266 T + p^{3} T^{2} \) |
| 89 | \( 1 - 250 T + p^{3} T^{2} \) |
| 97 | \( 1 - 322 T + p^{3} T^{2} \) |
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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
−9.270758423213359489836839633854, −8.724026629376574015164627077731, −7.85792526338978397365690881409, −6.61006376448070207958225075802, −5.78541359722552410299115163175, −4.84074491622510198242653176982, −4.10813986552442287122419321224, −2.90115787647007361230511012067, −1.51062609130605809440006413054, 0,
1.51062609130605809440006413054, 2.90115787647007361230511012067, 4.10813986552442287122419321224, 4.84074491622510198242653176982, 5.78541359722552410299115163175, 6.61006376448070207958225075802, 7.85792526338978397365690881409, 8.724026629376574015164627077731, 9.270758423213359489836839633854
