L(s) = 1 | − 2.74·3-s + 5-s − 2.35·7-s + 4.54·9-s − 0.675·11-s + 5.24·13-s − 2.74·15-s + 0.157·17-s + 3.38·19-s + 6.47·21-s − 7.83·23-s + 25-s − 4.24·27-s − 2.03·29-s − 2.89·31-s + 1.85·33-s − 2.35·35-s + 6.10·37-s − 14.3·39-s − 1.90·41-s + 9.82·43-s + 4.54·45-s + 12.3·47-s − 1.44·49-s − 0.431·51-s + 5.83·53-s − 0.675·55-s + ⋯ |
L(s) = 1 | − 1.58·3-s + 0.447·5-s − 0.890·7-s + 1.51·9-s − 0.203·11-s + 1.45·13-s − 0.709·15-s + 0.0380·17-s + 0.776·19-s + 1.41·21-s − 1.63·23-s + 0.200·25-s − 0.816·27-s − 0.378·29-s − 0.519·31-s + 0.322·33-s − 0.398·35-s + 1.00·37-s − 2.30·39-s − 0.297·41-s + 1.49·43-s + 0.677·45-s + 1.79·47-s − 0.206·49-s − 0.0604·51-s + 0.801·53-s − 0.0910·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9773003622\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9773003622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 0.675T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 0.157T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 + 7.83T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 5.83T + 53T^{2} \) |
| 59 | \( 1 + 0.257T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 3.95T + 97T^{2} \) |
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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
−7.60374983609703521031052187854, −6.94714294000490058765123861017, −6.07971122023427401339965639469, −5.93655942986183765247209346930, −5.39346096276670919166109084558, −4.28597940494577163349564023232, −3.74537659693836116059814682832, −2.64382537193785196679304412276, −1.45772307396289184472471000457, −0.56613778920385694545798038817,
0.56613778920385694545798038817, 1.45772307396289184472471000457, 2.64382537193785196679304412276, 3.74537659693836116059814682832, 4.28597940494577163349564023232, 5.39346096276670919166109084558, 5.93655942986183765247209346930, 6.07971122023427401339965639469, 6.94714294000490058765123861017, 7.60374983609703521031052187854