| L(s) = 1 | + 2-s + 3-s + 4-s + 0.542·5-s + 6-s + 3.05·7-s + 8-s + 9-s + 0.542·10-s + 0.632·11-s + 12-s + 13-s + 3.05·14-s + 0.542·15-s + 16-s + 2.75·17-s + 18-s + 2.05·19-s + 0.542·20-s + 3.05·21-s + 0.632·22-s + 6.49·23-s + 24-s − 4.70·25-s + 26-s + 27-s + 3.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.242·5-s + 0.408·6-s + 1.15·7-s + 0.353·8-s + 0.333·9-s + 0.171·10-s + 0.190·11-s + 0.288·12-s + 0.277·13-s + 0.816·14-s + 0.140·15-s + 0.250·16-s + 0.667·17-s + 0.235·18-s + 0.471·19-s + 0.121·20-s + 0.666·21-s + 0.134·22-s + 1.35·23-s + 0.204·24-s − 0.941·25-s + 0.196·26-s + 0.192·27-s + 0.577·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.749120218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.749120218\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 0.542T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 0.632T + 11T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 2.05T + 31T^{2} \) |
| 37 | \( 1 + 0.518T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 - 2.49T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 0.846T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 4.86T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
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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
−7.939593587322430587966672257911, −7.16124026537360449260469914121, −6.44024572221524509076160386752, −5.50571300573079553582870664779, −5.09104022267744527139926383701, −4.25505477656355754879549437579, −3.55091694304918479927647143379, −2.76770890403939738277659869425, −1.84307120409156599799924718698, −1.15650588154129988583079303193,
1.15650588154129988583079303193, 1.84307120409156599799924718698, 2.76770890403939738277659869425, 3.55091694304918479927647143379, 4.25505477656355754879549437579, 5.09104022267744527139926383701, 5.50571300573079553582870664779, 6.44024572221524509076160386752, 7.16124026537360449260469914121, 7.939593587322430587966672257911
