L(s) = 1 | + 1.34·3-s − 5-s − 7-s − 1.19·9-s + 1.02·11-s + 13-s − 1.34·15-s − 0.881·17-s + 3.90·19-s − 1.34·21-s + 0.975·23-s + 25-s − 5.63·27-s + 0.320·29-s − 4.46·31-s + 1.37·33-s + 35-s + 6.56·37-s + 1.34·39-s + 12.5·41-s + 3.71·43-s + 1.19·45-s + 5.12·47-s + 49-s − 1.18·51-s − 1.12·53-s − 1.02·55-s + ⋯ |
L(s) = 1 | + 0.776·3-s − 0.447·5-s − 0.377·7-s − 0.397·9-s + 0.308·11-s + 0.277·13-s − 0.347·15-s − 0.213·17-s + 0.896·19-s − 0.293·21-s + 0.203·23-s + 0.200·25-s − 1.08·27-s + 0.0594·29-s − 0.802·31-s + 0.239·33-s + 0.169·35-s + 1.07·37-s + 0.215·39-s + 1.95·41-s + 0.566·43-s + 0.177·45-s + 0.747·47-s + 0.142·49-s − 0.165·51-s − 0.154·53-s − 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.068384629\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.068384629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 11 | \( 1 - 1.02T + 11T^{2} \) |
| 17 | \( 1 + 0.881T + 17T^{2} \) |
| 19 | \( 1 - 3.90T + 19T^{2} \) |
| 23 | \( 1 - 0.975T + 23T^{2} \) |
| 29 | \( 1 - 0.320T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 0.430T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.616382774754011023684445436164, −7.71797126759826167416949298660, −7.37305101414734883142608373789, −6.26456396846595854186246949756, −5.65967566902144361904691826623, −4.55574463889217641490117909865, −3.72611943405535145678491323419, −3.06649679066555061899545824157, −2.21427354521559710151736097156, −0.805737653323568314788110739845,
0.805737653323568314788110739845, 2.21427354521559710151736097156, 3.06649679066555061899545824157, 3.72611943405535145678491323419, 4.55574463889217641490117909865, 5.65967566902144361904691826623, 6.26456396846595854186246949756, 7.37305101414734883142608373789, 7.71797126759826167416949298660, 8.616382774754011023684445436164