| L(s) = 1 | + 5-s + 2·13-s − 2·17-s + 2·19-s − 2·23-s + 25-s + 10·29-s − 10·31-s − 6·37-s + 6·41-s + 8·43-s + 8·47-s + 8·61-s + 2·65-s − 8·67-s − 2·73-s − 8·79-s + 4·83-s − 2·85-s − 2·89-s + 2·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.85·29-s − 1.79·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 1.02·61-s + 0.248·65-s − 0.977·67-s − 0.234·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s − 0.211·89-s + 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.583686256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583686256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.86107393406510, −14.23801044329537, −14.01910262309603, −13.38598611268376, −12.87661157157209, −12.32542069506414, −11.85246837759066, −11.13770540894474, −10.68047401360289, −10.24290168673191, −9.568615646190756, −8.942047154521998, −8.687211310111042, −7.884081564838343, −7.274257375804565, −6.780955706934108, −6.013694311196305, −5.680742279385598, −4.944415591126339, −4.255473007867888, −3.669011286460251, −2.868043755015633, −2.230087594447891, −1.443877468571452, −0.6181189540553150,
0.6181189540553150, 1.443877468571452, 2.230087594447891, 2.868043755015633, 3.669011286460251, 4.255473007867888, 4.944415591126339, 5.680742279385598, 6.013694311196305, 6.780955706934108, 7.274257375804565, 7.884081564838343, 8.687211310111042, 8.942047154521998, 9.568615646190756, 10.24290168673191, 10.68047401360289, 11.13770540894474, 11.85246837759066, 12.32542069506414, 12.87661157157209, 13.38598611268376, 14.01910262309603, 14.23801044329537, 14.86107393406510
