| L(s) = 1 | + 1.56·3-s − 5-s + 3.56·7-s − 0.561·9-s + 3.12·13-s − 1.56·15-s − 5.56·17-s + 2.43·19-s + 5.56·21-s − 7.12·23-s + 25-s − 5.56·27-s + 0.438·29-s − 8.68·31-s − 3.56·35-s + 9.80·37-s + 4.87·39-s + 10·41-s + 5.12·43-s + 0.561·45-s + 7.12·47-s + 5.68·49-s − 8.68·51-s − 4.43·53-s + 3.80·57-s + 13.3·59-s + 3.56·61-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.447·5-s + 1.34·7-s − 0.187·9-s + 0.866·13-s − 0.403·15-s − 1.34·17-s + 0.559·19-s + 1.21·21-s − 1.48·23-s + 0.200·25-s − 1.07·27-s + 0.0814·29-s − 1.55·31-s − 0.602·35-s + 1.61·37-s + 0.780·39-s + 1.56·41-s + 0.781·43-s + 0.0837·45-s + 1.03·47-s + 0.812·49-s − 1.21·51-s − 0.609·53-s + 0.504·57-s + 1.74·59-s + 0.456·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997323148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997323148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 0.876T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.71757776877214733658824471252, −7.41610614001904489919409690438, −6.21717514072273057621136537537, −5.70220538958208522503390394776, −4.73666520752694850221683402486, −4.07204203530136266060364144131, −3.57380954395202073141426774481, −2.39422471113254328376395343192, −2.00538506219869449645960073331, −0.78608357438396380565795427270,
0.78608357438396380565795427270, 2.00538506219869449645960073331, 2.39422471113254328376395343192, 3.57380954395202073141426774481, 4.07204203530136266060364144131, 4.73666520752694850221683402486, 5.70220538958208522503390394776, 6.21717514072273057621136537537, 7.41610614001904489919409690438, 7.71757776877214733658824471252
