| L(s) = 1 | + 3·3-s + 21.6·7-s + 9·9-s − 21.8·11-s − 70.6·13-s + 54.6·17-s − 76.0·19-s + 64.8·21-s + 89.0·23-s + 27·27-s − 51.8·29-s − 110.·31-s − 65.4·33-s − 9.56·37-s − 211.·39-s + 317.·41-s − 374.·43-s + 347.·47-s + 124.·49-s + 163.·51-s − 159.·53-s − 228.·57-s − 693.·59-s − 684.·61-s + 194.·63-s − 559.·67-s + 267.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.16·7-s + 0.333·9-s − 0.597·11-s − 1.50·13-s + 0.779·17-s − 0.918·19-s + 0.673·21-s + 0.807·23-s + 0.192·27-s − 0.331·29-s − 0.637·31-s − 0.345·33-s − 0.0425·37-s − 0.870·39-s + 1.20·41-s − 1.32·43-s + 1.07·47-s + 0.361·49-s + 0.450·51-s − 0.413·53-s − 0.530·57-s − 1.53·59-s − 1.43·61-s + 0.388·63-s − 1.02·67-s + 0.466·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 21.6T + 343T^{2} \) |
| 11 | \( 1 + 21.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 51.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.56T + 5.06e4T^{2} \) |
| 41 | \( 1 - 317.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 347.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 684.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 559.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 517.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 745.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 341.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 706.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 171.T + 9.12e5T^{2} \) |
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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
−8.024923201303592270426956947595, −7.68655838048480490719287168378, −6.94822049444735092822475414588, −5.71952359803259312130733733159, −4.92768681886135902456020369750, −4.36539581881456291816613764738, −3.12651838757813058459719523831, −2.30881927917321015947986167585, −1.43624977944600500256538622848, 0,
1.43624977944600500256538622848, 2.30881927917321015947986167585, 3.12651838757813058459719523831, 4.36539581881456291816613764738, 4.92768681886135902456020369750, 5.71952359803259312130733733159, 6.94822049444735092822475414588, 7.68655838048480490719287168378, 8.024923201303592270426956947595
