| L(s) = 1 | − 2.88·3-s + 0.781·5-s − 2.79·7-s + 5.34·9-s − 2.64·11-s − 5.63·13-s − 2.25·15-s − 4.71·17-s + 1.48·19-s + 8.06·21-s − 7.99·23-s − 4.38·25-s − 6.78·27-s − 2.41·29-s + 6.65·31-s + 7.63·33-s − 2.18·35-s − 2.28·37-s + 16.2·39-s − 3.37·41-s + 4.82·43-s + 4.17·45-s − 12.8·47-s + 0.797·49-s + 13.6·51-s + 4.78·53-s − 2.06·55-s + ⋯ |
| L(s) = 1 | − 1.66·3-s + 0.349·5-s − 1.05·7-s + 1.78·9-s − 0.796·11-s − 1.56·13-s − 0.582·15-s − 1.14·17-s + 0.339·19-s + 1.76·21-s − 1.66·23-s − 0.877·25-s − 1.30·27-s − 0.448·29-s + 1.19·31-s + 1.32·33-s − 0.368·35-s − 0.375·37-s + 2.60·39-s − 0.527·41-s + 0.735·43-s + 0.622·45-s − 1.87·47-s + 0.113·49-s + 1.90·51-s + 0.657·53-s − 0.278·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1122229305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1122229305\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 - 0.781T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 7.99T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 4.11T + 59T^{2} \) |
| 61 | \( 1 - 8.84T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 0.115T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 6.59T + 97T^{2} \) |
| 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
−8.374739728199092622228738351239, −7.39594013950522534281667855372, −6.84202245686245547408327587712, −6.09705694835240010883252915554, −5.63468533684466479082527358626, −4.82367130685060784680492249769, −4.16926718628333689136288288203, −2.84227779889386267983834079761, −1.89626426189304085922962389188, −0.19548801327518402444240695856,
0.19548801327518402444240695856, 1.89626426189304085922962389188, 2.84227779889386267983834079761, 4.16926718628333689136288288203, 4.82367130685060784680492249769, 5.63468533684466479082527358626, 6.09705694835240010883252915554, 6.84202245686245547408327587712, 7.39594013950522534281667855372, 8.374739728199092622228738351239
