L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 1.08·7-s − 8-s + 9-s − 10-s + 1.41·11-s − 12-s − 0.152·13-s − 1.08·14-s − 15-s + 16-s − 18-s + 2.82·19-s + 20-s − 1.08·21-s − 1.41·22-s − 6.67·23-s + 24-s + 25-s + 0.152·26-s − 27-s + 1.08·28-s + 1.81·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.409·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.426·11-s − 0.288·12-s − 0.0422·13-s − 0.289·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.648·19-s + 0.223·20-s − 0.236·21-s − 0.301·22-s − 1.39·23-s + 0.204·24-s + 0.200·25-s + 0.0298·26-s − 0.192·27-s + 0.204·28-s + 0.336·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 0.152T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 + 6.35T + 53T^{2} \) |
| 59 | \( 1 - 1.86T + 59T^{2} \) |
| 61 | \( 1 - 5.53T + 61T^{2} \) |
| 67 | \( 1 - 4.69T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 + 1.91T + 97T^{2} \) |
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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
−7.45994496002938394962104260754, −6.51802924012105617437874849752, −6.40676832925049401235831841282, −5.33543180590171475548146900924, −4.85509245440402473759467737670, −3.84428440091382285824615301294, −2.93745506782619521557693154165, −1.86482356516242484386640545554, −1.26046656338234364416289849151, 0,
1.26046656338234364416289849151, 1.86482356516242484386640545554, 2.93745506782619521557693154165, 3.84428440091382285824615301294, 4.85509245440402473759467737670, 5.33543180590171475548146900924, 6.40676832925049401235831841282, 6.51802924012105617437874849752, 7.45994496002938394962104260754