| L(s) = 1 | − 2-s + 3-s + 4-s + 2.15·5-s − 6-s − 1.48·7-s − 8-s + 9-s − 2.15·10-s − 11-s + 12-s + 5.53·13-s + 1.48·14-s + 2.15·15-s + 16-s + 2.33·17-s − 18-s + 3.42·19-s + 2.15·20-s − 1.48·21-s + 22-s + 3.87·23-s − 24-s − 0.335·25-s − 5.53·26-s + 27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.965·5-s − 0.408·6-s − 0.561·7-s − 0.353·8-s + 0.333·9-s − 0.682·10-s − 0.301·11-s + 0.288·12-s + 1.53·13-s + 0.397·14-s + 0.557·15-s + 0.250·16-s + 0.566·17-s − 0.235·18-s + 0.786·19-s + 0.482·20-s − 0.324·21-s + 0.213·22-s + 0.808·23-s − 0.204·24-s − 0.0671·25-s − 1.08·26-s + 0.192·27-s − 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.238436209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238436209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 2.15T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 - 2.33T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 - 8.37T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 + 5.39T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 + 4.59T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−8.667219569144808506492042114215, −7.80534094676543432651863204510, −7.10280018194882537504357990979, −6.19403915789551880514900041388, −5.82089261927039453660390188922, −4.71473970971267244223937090462, −3.42072649944591451556600467429, −2.96144650604747442980172507402, −1.82143010174598262334040082536, −0.988883588123450762761338848117,
0.988883588123450762761338848117, 1.82143010174598262334040082536, 2.96144650604747442980172507402, 3.42072649944591451556600467429, 4.71473970971267244223937090462, 5.82089261927039453660390188922, 6.19403915789551880514900041388, 7.10280018194882537504357990979, 7.80534094676543432651863204510, 8.667219569144808506492042114215
