L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 4·11-s + 2·13-s + 16-s − 6·17-s − 20-s + 4·22-s + 8·23-s + 25-s + 2·26-s − 10·29-s + 8·31-s + 32-s − 6·34-s + 2·37-s − 40-s − 2·41-s + 8·43-s + 4·44-s + 8·46-s + 4·47-s + 50-s + 2·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 1.85·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.158·40-s − 0.312·41-s + 1.21·43-s + 0.603·44-s + 1.17·46-s + 0.583·47-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.128897166\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.128897166\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.377615839643706529464870612175, −7.43248881648949058536574985753, −6.75409260319045124534751001266, −6.28177848330588507670744230995, −5.32517080585648699552867629168, −4.45300784628080382861352349832, −3.93540091151474404008857106243, −3.10457725835239419563641451463, −2.06591433643519364155542870836, −0.924219303462724660073974553477,
0.924219303462724660073974553477, 2.06591433643519364155542870836, 3.10457725835239419563641451463, 3.93540091151474404008857106243, 4.45300784628080382861352349832, 5.32517080585648699552867629168, 6.28177848330588507670744230995, 6.75409260319045124534751001266, 7.43248881648949058536574985753, 8.377615839643706529464870612175