L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 4·13-s + 14-s − 16-s + 4·19-s − 6·23-s + 4·26-s − 28-s + 6·31-s + 5·32-s − 2·37-s + 4·38-s + 2·41-s − 12·43-s − 6·46-s + 8·47-s + 49-s − 4·52-s − 14·53-s − 3·56-s − 6·59-s + 14·61-s + 6·62-s + 7·64-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.917·19-s − 1.25·23-s + 0.784·26-s − 0.188·28-s + 1.07·31-s + 0.883·32-s − 0.328·37-s + 0.648·38-s + 0.312·41-s − 1.82·43-s − 0.884·46-s + 1.16·47-s + 1/7·49-s − 0.554·52-s − 1.92·53-s − 0.400·56-s − 0.781·59-s + 1.79·61-s + 0.762·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.37253515328346, −13.07079858235694, −12.28597543916955, −12.13094322401124, −11.48637438353808, −11.24386259333891, −10.49132659206046, −9.993671819649953, −9.636557166278189, −9.030585450643326, −8.504342692362243, −8.187092915494398, −7.711259540558472, −6.983840131233791, −6.312081285870233, −6.092202282811523, −5.380557225016283, −5.091895191480936, −4.374581946823756, −4.041921604493336, −3.391343615238745, −3.043085655586491, −2.224177428084808, −1.487372282547842, −0.8646654880796494, 0,
0.8646654880796494, 1.487372282547842, 2.224177428084808, 3.043085655586491, 3.391343615238745, 4.041921604493336, 4.374581946823756, 5.091895191480936, 5.380557225016283, 6.092202282811523, 6.312081285870233, 6.983840131233791, 7.711259540558472, 8.187092915494398, 8.504342692362243, 9.030585450643326, 9.636557166278189, 9.993671819649953, 10.49132659206046, 11.24386259333891, 11.48637438353808, 12.13094322401124, 12.28597543916955, 13.07079858235694, 13.37253515328346