| L(s) = 1 | − 1.25·2-s − 0.414·4-s − 1.78·5-s + 3.04·8-s + 2.24·10-s + 1.78·11-s + 13-s − 3·16-s − 6.08·17-s + 0.737·20-s − 2.24·22-s + 6.08·23-s − 1.82·25-s − 1.25·26-s + 3.56·29-s − 10.8·31-s − 2.30·32-s + 7.65·34-s + 10.4·37-s − 5.41·40-s + 6.81·41-s − 9.65·43-s − 0.737·44-s − 7.65·46-s + 9.33·47-s + 2.30·50-s − 0.414·52-s + ⋯ |
| L(s) = 1 | − 0.890·2-s − 0.207·4-s − 0.796·5-s + 1.07·8-s + 0.709·10-s + 0.536·11-s + 0.277·13-s − 0.750·16-s − 1.47·17-s + 0.164·20-s − 0.478·22-s + 1.26·23-s − 0.365·25-s − 0.246·26-s + 0.661·29-s − 1.94·31-s − 0.407·32-s + 1.31·34-s + 1.72·37-s − 0.856·40-s + 1.06·41-s − 1.47·43-s − 0.111·44-s − 1.12·46-s + 1.36·47-s + 0.325·50-s − 0.0574·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 17 | \( 1 + 6.08T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 6.81T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 0.737T + 83T^{2} \) |
| 89 | \( 1 + 0.305T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−7.87338199158311645992065553117, −7.24592070258689366058027781010, −6.62866509238006501209398439315, −5.65105484398445220570392433312, −4.57556285470699569838177177025, −4.22875667623603840530898610636, −3.30061465426377501972362274650, −2.10342761118648762471649595660, −1.04374885829658419109018360402, 0,
1.04374885829658419109018360402, 2.10342761118648762471649595660, 3.30061465426377501972362274650, 4.22875667623603840530898610636, 4.57556285470699569838177177025, 5.65105484398445220570392433312, 6.62866509238006501209398439315, 7.24592070258689366058027781010, 7.87338199158311645992065553117
