L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 2·13-s − 14-s + 16-s + 17-s − 4·19-s + 2·20-s − 8·23-s − 25-s + 2·26-s − 28-s + 6·29-s + 32-s + 34-s − 2·35-s − 2·37-s − 4·38-s + 2·40-s + 10·41-s + 4·43-s − 8·46-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s + 0.609·43-s − 1.17·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.07006918388857, −12.67326760471423, −12.14244841099974, −11.91973582311733, −11.16266295530157, −10.70333081171653, −10.38531764948565, −9.851397632365338, −9.433684130479214, −8.959854041877804, −8.209426259939156, −8.012556105552876, −7.301018678477284, −6.702932232553740, −6.205310496507083, −5.978056242790270, −5.578119503748232, −4.870114390189891, −4.227001711687739, −4.000081859308686, −3.247932946753808, −2.682010418579596, −2.160153326981869, −1.665816474681244, −0.9386104699532511, 0,
0.9386104699532511, 1.665816474681244, 2.160153326981869, 2.682010418579596, 3.247932946753808, 4.000081859308686, 4.227001711687739, 4.870114390189891, 5.578119503748232, 5.978056242790270, 6.205310496507083, 6.702932232553740, 7.301018678477284, 8.012556105552876, 8.209426259939156, 8.959854041877804, 9.433684130479214, 9.851397632365338, 10.38531764948565, 10.70333081171653, 11.16266295530157, 11.91973582311733, 12.14244841099974, 12.67326760471423, 13.07006918388857