| L(s) = 1 | − 2-s + 3-s + 4-s − 3.86·5-s − 6-s + 7-s − 8-s + 9-s + 3.86·10-s + 2.24·11-s + 12-s − 14-s − 3.86·15-s + 16-s + 0.139·17-s − 18-s − 3.18·19-s − 3.86·20-s + 21-s − 2.24·22-s + 1.05·23-s − 24-s + 9.96·25-s + 27-s + 28-s − 6.34·29-s + 3.86·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.72·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.22·10-s + 0.676·11-s + 0.288·12-s − 0.267·14-s − 0.998·15-s + 0.250·16-s + 0.0337·17-s − 0.235·18-s − 0.731·19-s − 0.864·20-s + 0.218·21-s − 0.478·22-s + 0.219·23-s − 0.204·24-s + 1.99·25-s + 0.192·27-s + 0.188·28-s − 1.17·29-s + 0.706·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.86T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 17 | \( 1 - 0.139T + 17T^{2} \) |
| 19 | \( 1 + 3.18T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 8.90T + 53T^{2} \) |
| 59 | \( 1 - 0.140T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 7.74T + 97T^{2} \) |
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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
−7.84945487984906994162103269083, −6.99821836435162100298038396648, −6.68869628865085560072627733238, −5.42448566960035884321574053662, −4.46673176756165092006585365073, −3.85399333223870131118047716680, −3.25105074586197105558883762541, −2.19365717302361555780362427607, −1.15000836649445227555102086127, 0,
1.15000836649445227555102086127, 2.19365717302361555780362427607, 3.25105074586197105558883762541, 3.85399333223870131118047716680, 4.46673176756165092006585365073, 5.42448566960035884321574053662, 6.68869628865085560072627733238, 6.99821836435162100298038396648, 7.84945487984906994162103269083
