| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 11-s + 4·13-s − 4·14-s + 16-s − 17-s − 4·19-s + 22-s − 6·23-s − 5·25-s − 4·26-s + 4·28-s − 10·29-s − 8·31-s − 32-s + 34-s − 10·37-s + 4·38-s − 2·41-s + 8·43-s − 44-s + 6·46-s − 12·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.301·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s − 0.784·26-s + 0.755·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s − 1.64·37-s + 0.648·38-s − 0.312·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s − 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.211066988749483600419575981077, −7.76911583505060216630717646556, −6.96025876919790563640320344137, −5.91453504935033233075582405284, −5.40748508826916262368421830589, −4.26478959531665265800744020990, −3.57268117190856729752198720490, −1.96329227909637489505993202774, −1.72064219376073688263471934192, 0,
1.72064219376073688263471934192, 1.96329227909637489505993202774, 3.57268117190856729752198720490, 4.26478959531665265800744020990, 5.40748508826916262368421830589, 5.91453504935033233075582405284, 6.96025876919790563640320344137, 7.76911583505060216630717646556, 8.211066988749483600419575981077
