L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 5·13-s + 16-s − 6·17-s + 2·19-s − 22-s − 6·23-s − 5·25-s − 5·26-s − 3·29-s + 8·31-s − 32-s + 6·34-s + 2·37-s − 2·38-s + 6·41-s − 4·43-s + 44-s + 6·46-s − 6·47-s + 5·50-s + 5·52-s + 12·53-s + 3·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.213·22-s − 1.25·23-s − 25-s − 0.980·26-s − 0.557·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 0.884·46-s − 0.875·47-s + 0.707·50-s + 0.693·52-s + 1.64·53-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.46441474505048619894132918597, −6.58601578182622129138262694059, −6.18585667462119459133166735662, −5.52206766492078939423200691366, −4.32760430839758700665085405001, −3.90131164051747115843854234370, −2.87396906567531081875556593090, −2.00273212155307402658996612239, −1.20096769540244906903876784716, 0,
1.20096769540244906903876784716, 2.00273212155307402658996612239, 2.87396906567531081875556593090, 3.90131164051747115843854234370, 4.32760430839758700665085405001, 5.52206766492078939423200691366, 6.18585667462119459133166735662, 6.58601578182622129138262694059, 7.46441474505048619894132918597