| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s + 4·13-s + 3·14-s − 15-s + 16-s − 7·17-s − 2·18-s + 20-s − 3·21-s − 3·22-s − 6·23-s − 24-s + 25-s + 4·26-s + 5·27-s + 3·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s + 0.223·20-s − 0.654·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.962·27-s + 0.566·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.837318408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.837318408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.366185869663910773564503815141, −7.78868583277833887437993467032, −6.61378644987028618357181095553, −6.10946402786816038690191989043, −5.51542178024399932948913909686, −4.64198156805021315941819752698, −4.23449851555026021414871716821, −2.77488811174550060171323193305, −2.20830561425569631317168057215, −0.905014131053380638466907402001,
0.905014131053380638466907402001, 2.20830561425569631317168057215, 2.77488811174550060171323193305, 4.23449851555026021414871716821, 4.64198156805021315941819752698, 5.51542178024399932948913909686, 6.10946402786816038690191989043, 6.61378644987028618357181095553, 7.78868583277833887437993467032, 8.366185869663910773564503815141
