| L(s) = 1 | + 9·3-s − 25·5-s + 188·7-s + 81·9-s + 121·11-s + 698·13-s − 225·15-s + 1.89e3·17-s − 2.42e3·19-s + 1.69e3·21-s + 2.34e3·23-s + 625·25-s + 729·27-s + 990·29-s + 128·31-s + 1.08e3·33-s − 4.70e3·35-s − 3.20e3·37-s + 6.28e3·39-s + 1.73e4·41-s − 6.65e3·43-s − 2.02e3·45-s − 2.50e4·47-s + 1.85e4·49-s + 1.70e4·51-s − 1.82e4·53-s − 3.02e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.45·7-s + 1/3·9-s + 0.301·11-s + 1.14·13-s − 0.258·15-s + 1.58·17-s − 1.54·19-s + 0.837·21-s + 0.922·23-s + 1/5·25-s + 0.192·27-s + 0.218·29-s + 0.0239·31-s + 0.174·33-s − 0.648·35-s − 0.384·37-s + 0.661·39-s + 1.61·41-s − 0.548·43-s − 0.149·45-s − 1.65·47-s + 1.10·49-s + 0.915·51-s − 0.892·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.684456021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.684456021\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 188 T + p^{5} T^{2} \) |
| 13 | \( 1 - 698 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1890 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2428 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2340 T + p^{5} T^{2} \) |
| 29 | \( 1 - 990 T + p^{5} T^{2} \) |
| 31 | \( 1 - 128 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3202 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17370 T + p^{5} T^{2} \) |
| 43 | \( 1 + 6652 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25020 T + p^{5} T^{2} \) |
| 53 | \( 1 + 18246 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8652 T + p^{5} T^{2} \) |
| 61 | \( 1 - 37682 T + p^{5} T^{2} \) |
| 67 | \( 1 + 18676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 2340 T + p^{5} T^{2} \) |
| 73 | \( 1 - 43058 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65300 T + p^{5} T^{2} \) |
| 83 | \( 1 + 55308 T + p^{5} T^{2} \) |
| 89 | \( 1 + 64806 T + p^{5} T^{2} \) |
| 97 | \( 1 - 38306 T + p^{5} T^{2} \) |
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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
−9.689931215296000954282837832008, −8.513981243848699639414765742293, −8.269827549253561423024400083523, −7.35567634279650824260029021365, −6.22541400457682266545682749625, −5.04579607711984970655295089809, −4.15616509142146574176897425504, −3.21913728522344074149345842608, −1.82309285761218339648562654930, −0.968559901238569497163450775928,
0.968559901238569497163450775928, 1.82309285761218339648562654930, 3.21913728522344074149345842608, 4.15616509142146574176897425504, 5.04579607711984970655295089809, 6.22541400457682266545682749625, 7.35567634279650824260029021365, 8.269827549253561423024400083523, 8.513981243848699639414765742293, 9.689931215296000954282837832008
