| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 4.82·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 9.65·10-s + 43.4·11-s + 12·12-s + 13·13-s + 14·14-s + 14.4·15-s + 16·16-s − 101.·17-s + 18·18-s + 36.8·19-s + 19.3·20-s + 21·21-s + 86.8·22-s + 209.·23-s + 24·24-s − 101.·25-s + 26·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.431·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.305·10-s + 1.18·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.249·15-s + 0.250·16-s − 1.44·17-s + 0.235·18-s + 0.445·19-s + 0.215·20-s + 0.218·21-s + 0.841·22-s + 1.90·23-s + 0.204·24-s − 0.813·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.698514190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.698514190\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 4.82T + 125T^{2} \) |
| 11 | \( 1 - 43.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 209.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 93.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 509.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 245.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 719.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 643.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 405.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 155.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 817.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 572.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 182.T + 9.12e5T^{2} \) |
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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
−10.56973830652370495346147219693, −9.235225557180542828345413497243, −8.886034256432422335795781298607, −7.49911965404543354648502227437, −6.73737349372174431165645018632, −5.73585873643278905628316881748, −4.58080287783492193337600005361, −3.71016219246673433024269128122, −2.45816906165170283870221694042, −1.34067712607109347442410613515,
1.34067712607109347442410613515, 2.45816906165170283870221694042, 3.71016219246673433024269128122, 4.58080287783492193337600005361, 5.73585873643278905628316881748, 6.73737349372174431165645018632, 7.49911965404543354648502227437, 8.886034256432422335795781298607, 9.235225557180542828345413497243, 10.56973830652370495346147219693
