| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 13-s + 15-s − 6·17-s + 4·21-s + 4·23-s + 25-s + 27-s + 6·29-s + 8·31-s + 4·35-s − 2·37-s − 39-s − 10·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 6·51-s − 2·53-s − 4·59-s − 14·61-s + 4·63-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s − 0.520·59-s − 1.79·61-s + 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.950165247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.950165247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.31818412842289, −12.13238185940268, −11.51716868037763, −11.08679046634335, −10.73154920217114, −10.27639383002705, −9.635761037414156, −9.396982222263260, −8.572262444285715, −8.463100335630754, −8.127552465981146, −7.521737384946521, −6.833628746902947, −6.653304376635460, −6.102872801718564, −5.209802083032287, −4.882457480223036, −4.663801412133477, −4.093449149285117, −3.252270937291208, −2.881811880939910, −2.148590888740002, −1.846809033598165, −1.269007038178571, −0.5398013990424200,
0.5398013990424200, 1.269007038178571, 1.846809033598165, 2.148590888740002, 2.881811880939910, 3.252270937291208, 4.093449149285117, 4.663801412133477, 4.882457480223036, 5.209802083032287, 6.102872801718564, 6.653304376635460, 6.833628746902947, 7.521737384946521, 8.127552465981146, 8.463100335630754, 8.572262444285715, 9.396982222263260, 9.635761037414156, 10.27639383002705, 10.73154920217114, 11.08679046634335, 11.51716868037763, 12.13238185940268, 12.31818412842289
