L(s) = 1 | + 0.932·3-s + 3.17·5-s − 2.52·7-s − 2.13·9-s − 13-s + 2.95·15-s − 2.83·17-s + 8.68·19-s − 2.35·21-s + 5.07·23-s + 5.07·25-s − 4.78·27-s − 0.407·29-s + 4.04·31-s − 8.01·35-s + 9.08·37-s − 0.932·39-s + 2.86·41-s − 4.20·43-s − 6.76·45-s + 1.30·47-s − 0.627·49-s − 2.64·51-s − 0.552·53-s + 8.09·57-s + 10.8·59-s + 2.01·61-s + ⋯ |
L(s) = 1 | + 0.538·3-s + 1.41·5-s − 0.954·7-s − 0.710·9-s − 0.277·13-s + 0.764·15-s − 0.687·17-s + 1.99·19-s − 0.513·21-s + 1.05·23-s + 1.01·25-s − 0.920·27-s − 0.0756·29-s + 0.727·31-s − 1.35·35-s + 1.49·37-s − 0.149·39-s + 0.447·41-s − 0.641·43-s − 1.00·45-s + 0.190·47-s − 0.0896·49-s − 0.370·51-s − 0.0758·53-s + 1.07·57-s + 1.40·59-s + 0.258·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.781451958\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.781451958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.932T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 8.68T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 + 0.407T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 - 9.08T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 0.552T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 + 7.43T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 0.651T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.095731093125546609927277841345, −7.25223062930656265795693118918, −6.59259953002846987365291973762, −5.84693837381427442192644567099, −5.41642509170715216220116761295, −4.47507561558583901397738348810, −3.16833168105210193971266271610, −2.90421271932591421077788785030, −2.02821417774770554054197515914, −0.841294778313716946293379150184,
0.841294778313716946293379150184, 2.02821417774770554054197515914, 2.90421271932591421077788785030, 3.16833168105210193971266271610, 4.47507561558583901397738348810, 5.41642509170715216220116761295, 5.84693837381427442192644567099, 6.59259953002846987365291973762, 7.25223062930656265795693118918, 8.095731093125546609927277841345