| L(s) = 1 | − 2.50·3-s + 0.621·5-s + 1.81·7-s + 3.27·9-s − 0.483·11-s − 2.34·13-s − 1.55·15-s − 4.38·17-s + 5.17·19-s − 4.54·21-s − 7.24·23-s − 4.61·25-s − 0.688·27-s − 5.05·29-s − 4.08·31-s + 1.21·33-s + 1.12·35-s + 1.40·37-s + 5.87·39-s + 7.17·41-s − 4.88·43-s + 2.03·45-s + 6.60·47-s − 3.71·49-s + 10.9·51-s − 2.78·53-s − 0.300·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 0.277·5-s + 0.685·7-s + 1.09·9-s − 0.145·11-s − 0.650·13-s − 0.401·15-s − 1.06·17-s + 1.18·19-s − 0.991·21-s − 1.51·23-s − 0.922·25-s − 0.132·27-s − 0.938·29-s − 0.734·31-s + 0.210·33-s + 0.190·35-s + 0.230·37-s + 0.940·39-s + 1.12·41-s − 0.745·43-s + 0.303·45-s + 0.964·47-s − 0.530·49-s + 1.53·51-s − 0.382·53-s − 0.0405·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8228050587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8228050587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.621T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 + 0.483T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 7.17T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.98T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 - 7.88T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66455594344058180678026266653, −7.08826979091280636478039524006, −6.30677358217225129405630556294, −5.59726330785358024520938645493, −5.28696606862412116795616869379, −4.46063222931366786826691516783, −3.78930737640037682241562794153, −2.40885450170531358644806526745, −1.69012136254787955336631057478, −0.48030679813806352697478756163,
0.48030679813806352697478756163, 1.69012136254787955336631057478, 2.40885450170531358644806526745, 3.78930737640037682241562794153, 4.46063222931366786826691516783, 5.28696606862412116795616869379, 5.59726330785358024520938645493, 6.30677358217225129405630556294, 7.08826979091280636478039524006, 7.66455594344058180678026266653
