| L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s + 5·13-s + 16-s − 6·17-s − 4·19-s − 6·22-s − 6·23-s − 5·25-s + 5·26-s − 6·29-s − 31-s + 32-s − 6·34-s − 37-s − 4·38-s + 6·41-s − 43-s − 6·44-s − 6·46-s + 6·47-s − 5·50-s + 5·52-s + 6·53-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.27·22-s − 1.25·23-s − 25-s + 0.980·26-s − 1.11·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 0.164·37-s − 0.648·38-s + 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.884·46-s + 0.875·47-s − 0.707·50-s + 0.693·52-s + 0.824·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.322567467361658853622879025591, −7.75571829894035808861745121169, −6.83258442473794854766223031310, −5.91864842852306924737657425731, −5.53634730988965039358149929804, −4.34596400781076763888194778584, −3.84967222428535684217921793847, −2.60629849220782543841100988768, −1.91689542680922239943101317453, 0,
1.91689542680922239943101317453, 2.60629849220782543841100988768, 3.84967222428535684217921793847, 4.34596400781076763888194778584, 5.53634730988965039358149929804, 5.91864842852306924737657425731, 6.83258442473794854766223031310, 7.75571829894035808861745121169, 8.322567467361658853622879025591
