| L(s) = 1 | + 2-s − 0.644·3-s + 4-s − 2.05·5-s − 0.644·6-s + 8-s − 2.58·9-s − 2.05·10-s + 4.67·11-s − 0.644·12-s + 3.01·13-s + 1.32·15-s + 16-s − 5.19·17-s − 2.58·18-s − 2.70·19-s − 2.05·20-s + 4.67·22-s + 5.34·23-s − 0.644·24-s − 0.762·25-s + 3.01·26-s + 3.59·27-s − 29-s + 1.32·30-s + 3.59·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.371·3-s + 0.5·4-s − 0.920·5-s − 0.263·6-s + 0.353·8-s − 0.861·9-s − 0.650·10-s + 1.40·11-s − 0.185·12-s + 0.835·13-s + 0.342·15-s + 0.250·16-s − 1.26·17-s − 0.609·18-s − 0.620·19-s − 0.460·20-s + 0.996·22-s + 1.11·23-s − 0.131·24-s − 0.152·25-s + 0.590·26-s + 0.692·27-s − 0.185·29-s + 0.242·30-s + 0.646·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053182284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053182284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.644T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 0.461T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 0.673T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.41T + 79T^{2} \) |
| 83 | \( 1 - 0.461T + 83T^{2} \) |
| 89 | \( 1 - 0.827T + 89T^{2} \) |
| 97 | \( 1 + 0.712T + 97T^{2} \) |
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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
−8.760126001795501388571386526487, −8.045256510201506668821029965736, −6.97838669466560416373230650786, −6.46633289306596699833393977558, −5.78901180768493114125173524339, −4.73029714393375690341609119450, −4.05235642476626868692892516767, −3.40806610578006005827783985428, −2.25519050548202502953563441813, −0.813177630844367834486084504674,
0.813177630844367834486084504674, 2.25519050548202502953563441813, 3.40806610578006005827783985428, 4.05235642476626868692892516767, 4.73029714393375690341609119450, 5.78901180768493114125173524339, 6.46633289306596699833393977558, 6.97838669466560416373230650786, 8.045256510201506668821029965736, 8.760126001795501388571386526487
