L(s) = 1 | + 2-s + 2.67·3-s + 4-s − 5-s + 2.67·6-s − 7-s + 8-s + 4.16·9-s − 10-s − 5.31·11-s + 2.67·12-s − 6.03·13-s − 14-s − 2.67·15-s + 16-s − 1.97·17-s + 4.16·18-s − 0.363·19-s − 20-s − 2.67·21-s − 5.31·22-s − 3.91·23-s + 2.67·24-s + 25-s − 6.03·26-s + 3.10·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.54·3-s + 0.5·4-s − 0.447·5-s + 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.38·9-s − 0.316·10-s − 1.60·11-s + 0.772·12-s − 1.67·13-s − 0.267·14-s − 0.690·15-s + 0.250·16-s − 0.478·17-s + 0.980·18-s − 0.0834·19-s − 0.223·20-s − 0.583·21-s − 1.13·22-s − 0.817·23-s + 0.546·24-s + 0.200·25-s − 1.18·26-s + 0.597·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 0.363T + 19T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 3.94T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 73 | \( 1 - 0.291T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 9.59T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.81695190809150945664650700557, −7.39975283840419919412994050400, −6.62747570186612852473612454312, −5.49360343777022122981421432317, −4.75901952913142731578796940352, −4.07263682670243197600297259536, −3.12409186667214147218670368294, −2.63039209048203690438045535468, −2.01573020617187281406291729892, 0,
2.01573020617187281406291729892, 2.63039209048203690438045535468, 3.12409186667214147218670368294, 4.07263682670243197600297259536, 4.75901952913142731578796940352, 5.49360343777022122981421432317, 6.62747570186612852473612454312, 7.39975283840419919412994050400, 7.81695190809150945664650700557