| L(s) = 1 | − 2·4-s + 4.35·5-s − 3·7-s − 4.35·11-s + 4·16-s + 4.35·17-s − 8.71·20-s − 8.71·23-s + 14.0·25-s + 6·28-s − 13.0·35-s − 43-s + 8.71·44-s − 4.35·47-s + 2·49-s − 19.0·55-s − 15·61-s − 8·64-s − 8.71·68-s − 11·73-s + 13.0·77-s + 17.4·80-s − 8.71·83-s + 19.0·85-s + 17.4·92-s − 28.0·100-s − 17.4·101-s + ⋯ |
| L(s) = 1 | − 4-s + 1.94·5-s − 1.13·7-s − 1.31·11-s + 16-s + 1.05·17-s − 1.94·20-s − 1.81·23-s + 2.80·25-s + 1.13·28-s − 2.21·35-s − 0.152·43-s + 1.31·44-s − 0.635·47-s + 0.285·49-s − 2.56·55-s − 1.92·61-s − 64-s − 1.05·68-s − 1.28·73-s + 1.49·77-s + 1.94·80-s − 0.956·83-s + 2.06·85-s + 1.81·92-s − 2.80·100-s − 1.73·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 4.35T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 4.35T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.400977761572664544871474452177, −7.64506286107540222858928901865, −6.50631274564408047293057032590, −5.79275455165857674966309899785, −5.49541877642885423146960506516, −4.56167508086412660153038421343, −3.33953650809609433802311354421, −2.62334504861051455194550320989, −1.49618975806630861241107453494, 0,
1.49618975806630861241107453494, 2.62334504861051455194550320989, 3.33953650809609433802311354421, 4.56167508086412660153038421343, 5.49541877642885423146960506516, 5.79275455165857674966309899785, 6.50631274564408047293057032590, 7.64506286107540222858928901865, 8.400977761572664544871474452177
