| L(s) = 1 | + 3.30·3-s − 1.63·7-s + 7.89·9-s − 4.67·11-s + 4.75·13-s + 1.41·17-s + 19-s − 5.38·21-s + 1.96·23-s + 16.1·27-s + 6.85·29-s + 5.08·31-s − 15.4·33-s − 10.6·37-s + 15.6·39-s − 4.52·41-s + 7.83·43-s + 10.9·47-s − 4.34·49-s + 4.66·51-s + 1.55·53-s + 3.30·57-s − 6.81·59-s − 0.109·61-s − 12.8·63-s + 10.5·67-s + 6.48·69-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.616·7-s + 2.63·9-s − 1.40·11-s + 1.31·13-s + 0.343·17-s + 0.229·19-s − 1.17·21-s + 0.409·23-s + 3.11·27-s + 1.27·29-s + 0.912·31-s − 2.68·33-s − 1.74·37-s + 2.51·39-s − 0.706·41-s + 1.19·43-s + 1.59·47-s − 0.620·49-s + 0.653·51-s + 0.214·53-s + 0.437·57-s − 0.887·59-s − 0.0139·61-s − 1.62·63-s + 1.29·67-s + 0.780·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.907746836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.907746836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 + 0.109T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.519T + 73T^{2} \) |
| 79 | \( 1 - 0.840T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 7.44T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−8.459147048792876735933323851715, −8.014876437056643078219750867833, −7.21652858518271891427818910671, −6.53828159139984104330271052815, −5.43965244284526389067980493797, −4.45040997678695802033102514524, −3.54029383780537204278981138292, −3.01961105908707122721732336403, −2.30558340894034523111948461928, −1.11192756158897836347224744797,
1.11192756158897836347224744797, 2.30558340894034523111948461928, 3.01961105908707122721732336403, 3.54029383780537204278981138292, 4.45040997678695802033102514524, 5.43965244284526389067980493797, 6.53828159139984104330271052815, 7.21652858518271891427818910671, 8.014876437056643078219750867833, 8.459147048792876735933323851715
