L(s) = 1 | + 2-s + 1.72·3-s + 4-s + 2.42·5-s + 1.72·6-s + 8-s − 0.0331·9-s + 2.42·10-s + 3.72·11-s + 1.72·12-s + 6.58·13-s + 4.17·15-s + 16-s − 17-s − 0.0331·18-s − 5.58·19-s + 2.42·20-s + 3.72·22-s − 5.97·23-s + 1.72·24-s + 0.862·25-s + 6.58·26-s − 5.22·27-s − 0.772·29-s + 4.17·30-s − 9.61·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.994·3-s + 0.5·4-s + 1.08·5-s + 0.703·6-s + 0.353·8-s − 0.0110·9-s + 0.765·10-s + 1.12·11-s + 0.497·12-s + 1.82·13-s + 1.07·15-s + 0.250·16-s − 0.242·17-s − 0.00780·18-s − 1.28·19-s + 0.541·20-s + 0.793·22-s − 1.24·23-s + 0.351·24-s + 0.172·25-s + 1.29·26-s − 1.00·27-s − 0.143·29-s + 0.761·30-s − 1.72·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.643209776\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.643209776\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 + 0.772T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 6.25T + 43T^{2} \) |
| 47 | \( 1 + 0.671T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 - 1.61T + 73T^{2} \) |
| 79 | \( 1 - 4.56T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 7.54T + 97T^{2} \) |
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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
−9.194919715848535310359639445175, −8.671951270463075274061334906214, −7.933873406458413616616302847468, −6.58366175810800704954978606235, −6.21266436232297632747263610300, −5.39839391469597585550072534483, −3.91499165232124486898813653757, −3.65649157915110251083363834346, −2.24775257255724354123721675843, −1.65739864992939336040012920317,
1.65739864992939336040012920317, 2.24775257255724354123721675843, 3.65649157915110251083363834346, 3.91499165232124486898813653757, 5.39839391469597585550072534483, 6.21266436232297632747263610300, 6.58366175810800704954978606235, 7.933873406458413616616302847468, 8.671951270463075274061334906214, 9.194919715848535310359639445175