| L(s) = 1 | − 2·4-s − 5-s + 11-s + 4·13-s + 4·16-s + 2·17-s + 6·19-s + 2·20-s + 5·23-s − 4·25-s − 10·29-s − 31-s − 5·37-s − 2·41-s − 8·43-s − 2·44-s + 8·47-s − 8·52-s + 6·53-s − 55-s + 3·59-s + 2·61-s − 8·64-s − 4·65-s − 3·67-s − 4·68-s − 71-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.301·11-s + 1.10·13-s + 16-s + 0.485·17-s + 1.37·19-s + 0.447·20-s + 1.04·23-s − 4/5·25-s − 1.85·29-s − 0.179·31-s − 0.821·37-s − 0.312·41-s − 1.21·43-s − 0.301·44-s + 1.16·47-s − 1.10·52-s + 0.824·53-s − 0.134·55-s + 0.390·59-s + 0.256·61-s − 64-s − 0.496·65-s − 0.366·67-s − 0.485·68-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396499075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396499075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.379217653625850464982927035282, −7.56983916575516165617607085292, −7.03931028753471523628290962944, −5.78934185310610905216235400631, −5.45325947409667105532816166410, −4.50496459088241662864741392768, −3.58696683503113070124091225450, −3.36085044189202617570655653362, −1.67253698278596207710749593695, −0.69132514251150307357091373163,
0.69132514251150307357091373163, 1.67253698278596207710749593695, 3.36085044189202617570655653362, 3.58696683503113070124091225450, 4.50496459088241662864741392768, 5.45325947409667105532816166410, 5.78934185310610905216235400631, 7.03931028753471523628290962944, 7.56983916575516165617607085292, 8.379217653625850464982927035282
