| L(s) = 1 | + 5·5-s + 22.8·7-s − 11.0·11-s − 11.6·13-s + 10.0·17-s − 117.·19-s − 172.·23-s + 25·25-s + 178.·29-s − 140.·31-s + 114.·35-s + 250.·37-s + 361.·41-s + 360.·43-s + 600.·47-s + 181.·49-s + 201.·53-s − 55.4·55-s − 415.·59-s − 54.6·61-s − 58.1·65-s + 531.·67-s + 933.·71-s − 560.·73-s − 253.·77-s − 810.·79-s − 538.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.23·7-s − 0.303·11-s − 0.248·13-s + 0.143·17-s − 1.42·19-s − 1.56·23-s + 0.200·25-s + 1.14·29-s − 0.814·31-s + 0.552·35-s + 1.11·37-s + 1.37·41-s + 1.27·43-s + 1.86·47-s + 0.528·49-s + 0.521·53-s − 0.135·55-s − 0.917·59-s − 0.114·61-s − 0.111·65-s + 0.968·67-s + 1.56·71-s − 0.898·73-s − 0.375·77-s − 1.15·79-s − 0.711·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.668835434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.668835434\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 22.8T + 343T^{2} \) |
| 11 | \( 1 + 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 560.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 686.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−8.633606732102824725436919502426, −7.970627532516751309328718942869, −7.34113304741699661952286298249, −6.18203592365138161002893201068, −5.65410904067746767851249594977, −4.59140837163798799139414204620, −4.08685069583425455750513953733, −2.53931372010116937766953503282, −1.95647970699696974840266089680, −0.74086958051538614133509596372,
0.74086958051538614133509596372, 1.95647970699696974840266089680, 2.53931372010116937766953503282, 4.08685069583425455750513953733, 4.59140837163798799139414204620, 5.65410904067746767851249594977, 6.18203592365138161002893201068, 7.34113304741699661952286298249, 7.970627532516751309328718942869, 8.633606732102824725436919502426
