| L(s) = 1 | − 4.72·2-s + 14.3·4-s + 17.6·7-s − 29.9·8-s − 34.2·11-s + 53.8·13-s − 83.3·14-s + 26.9·16-s + 74.7·17-s − 89.5·19-s + 161.·22-s − 176.·23-s − 254.·26-s + 252.·28-s − 194.·29-s + 107.·31-s + 112.·32-s − 353.·34-s − 430.·37-s + 423.·38-s + 108.·41-s + 409.·43-s − 490.·44-s + 832.·46-s − 409.·47-s − 32.3·49-s + 772.·52-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.79·4-s + 0.951·7-s − 1.32·8-s − 0.938·11-s + 1.14·13-s − 1.59·14-s + 0.420·16-s + 1.06·17-s − 1.08·19-s + 1.56·22-s − 1.59·23-s − 1.92·26-s + 1.70·28-s − 1.24·29-s + 0.625·31-s + 0.621·32-s − 1.78·34-s − 1.91·37-s + 1.80·38-s + 0.414·41-s + 1.45·43-s − 1.68·44-s + 2.66·46-s − 1.26·47-s − 0.0942·49-s + 2.06·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.72T + 8T^{2} \) |
| 7 | \( 1 - 17.6T + 343T^{2} \) |
| 11 | \( 1 + 34.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 430.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 409.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 409.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 24.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 295.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 305.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 915.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 228.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 158.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 319.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 936.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 920.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 914.T + 9.12e5T^{2} \) |
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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
−9.698155752931631904776275972087, −8.565942499392721313775685281859, −8.165407489051511398411527781626, −7.50432096779957889607747771894, −6.33604589010960908119855121652, −5.33419839903511723584942004845, −3.87076781099840177216705696126, −2.26837266469943916964161671937, −1.36847008875610006275517218067, 0,
1.36847008875610006275517218067, 2.26837266469943916964161671937, 3.87076781099840177216705696126, 5.33419839903511723584942004845, 6.33604589010960908119855121652, 7.50432096779957889607747771894, 8.165407489051511398411527781626, 8.565942499392721313775685281859, 9.698155752931631904776275972087
