| L(s) = 1 | + 2.08·2-s − 0.777·3-s + 2.35·4-s − 3.75·5-s − 1.62·6-s − 3.38·7-s + 0.741·8-s − 2.39·9-s − 7.84·10-s − 6.13·11-s − 1.83·12-s − 7.07·14-s + 2.92·15-s − 3.16·16-s + 5.82·17-s − 4.99·18-s − 3.74·19-s − 8.85·20-s + 2.63·21-s − 12.7·22-s − 2.96·23-s − 0.576·24-s + 9.13·25-s + 4.19·27-s − 7.98·28-s − 3.91·29-s + 6.09·30-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.448·3-s + 1.17·4-s − 1.68·5-s − 0.662·6-s − 1.28·7-s + 0.262·8-s − 0.798·9-s − 2.48·10-s − 1.84·11-s − 0.528·12-s − 1.89·14-s + 0.754·15-s − 0.790·16-s + 1.41·17-s − 1.17·18-s − 0.858·19-s − 1.98·20-s + 0.574·21-s − 2.72·22-s − 0.617·23-s − 0.117·24-s + 1.82·25-s + 0.807·27-s − 1.50·28-s − 0.726·29-s + 1.11·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3069988148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3069988148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 + 0.777T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 37 | \( 1 + 0.194T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 6.93T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 + 0.957T + 73T^{2} \) |
| 79 | \( 1 - 3.67T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−7.889158281323227799362743917322, −7.47175229654450611824598485631, −6.44962361417919626843648382821, −5.90797453214405384986188340377, −5.21446978800885158949977236186, −4.55675709543020440802802907428, −3.61315043191870535970420460381, −3.21923253397127402421055645537, −2.55090640594350165141693289294, −0.22240796076348272827777745545,
0.22240796076348272827777745545, 2.55090640594350165141693289294, 3.21923253397127402421055645537, 3.61315043191870535970420460381, 4.55675709543020440802802907428, 5.21446978800885158949977236186, 5.90797453214405384986188340377, 6.44962361417919626843648382821, 7.47175229654450611824598485631, 7.889158281323227799362743917322
