| L(s) = 1 | − 4.76·2-s − 2.80·3-s + 14.6·4-s − 15.6·5-s + 13.3·6-s + 31.0·7-s − 31.7·8-s − 19.1·9-s + 74.5·10-s − 41.1·12-s − 22.0·13-s − 147.·14-s + 43.8·15-s + 33.8·16-s + 17·17-s + 91.1·18-s − 124.·19-s − 229.·20-s − 86.9·21-s − 171.·23-s + 89.0·24-s + 119.·25-s + 104.·26-s + 129.·27-s + 455.·28-s − 229.·29-s − 208.·30-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.539·3-s + 1.83·4-s − 1.39·5-s + 0.908·6-s + 1.67·7-s − 1.40·8-s − 0.708·9-s + 2.35·10-s − 0.989·12-s − 0.469·13-s − 2.81·14-s + 0.755·15-s + 0.529·16-s + 0.242·17-s + 1.19·18-s − 1.50·19-s − 2.56·20-s − 0.903·21-s − 1.55·23-s + 0.757·24-s + 0.959·25-s + 0.790·26-s + 0.921·27-s + 3.07·28-s − 1.46·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06914472415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06914472415\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.76T + 8T^{2} \) |
| 3 | \( 1 + 2.80T + 27T^{2} \) |
| 5 | \( 1 + 15.6T + 125T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 13 | \( 1 + 22.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 580.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 207.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 387.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 252.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 272.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 730.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 444.T + 9.12e5T^{2} \) |
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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
−8.424036126117563336343266728941, −8.162015289513068286447688513389, −7.64835490907074967289917127190, −6.81559109208900950454532779613, −5.74639521797619203583076459096, −4.73013869366598736934522545092, −3.89136177147784515395070896606, −2.36085723961675765366145138162, −1.49901268562894312684442289950, −0.16303921856821007386382857675,
0.16303921856821007386382857675, 1.49901268562894312684442289950, 2.36085723961675765366145138162, 3.89136177147784515395070896606, 4.73013869366598736934522545092, 5.74639521797619203583076459096, 6.81559109208900950454532779613, 7.64835490907074967289917127190, 8.162015289513068286447688513389, 8.424036126117563336343266728941
