| L(s) = 1 | + 4.47·7-s − 3·9-s + 2·11-s + 4.47·13-s − 6·19-s − 4.47·23-s + 4.47·37-s − 2·41-s + 13.4·47-s + 13.0·49-s + 13.4·53-s + 14·59-s − 13.4·63-s + 8.94·77-s + 9·81-s − 14·89-s + 20.0·91-s − 6·99-s − 4.47·103-s − 13.4·117-s + ⋯ |
| L(s) = 1 | + 1.69·7-s − 9-s + 0.603·11-s + 1.24·13-s − 1.37·19-s − 0.932·23-s + 0.735·37-s − 0.312·41-s + 1.95·47-s + 1.85·49-s + 1.84·53-s + 1.82·59-s − 1.69·63-s + 1.01·77-s + 81-s − 1.48·89-s + 2.09·91-s − 0.603·99-s − 0.440·103-s − 1.24·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.497932320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497932320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.315712647209521079979155237226, −7.43856900649409982691375733078, −6.52104739049924581790185829593, −5.81869608323955159768823193478, −5.30717235200283562553062365044, −4.17022715486239219539841764160, −3.95877098433259406134766107408, −2.56389319830202201244705472492, −1.85309766336570377699208936780, −0.850970556049532310052870196815,
0.850970556049532310052870196815, 1.85309766336570377699208936780, 2.56389319830202201244705472492, 3.95877098433259406134766107408, 4.17022715486239219539841764160, 5.30717235200283562553062365044, 5.81869608323955159768823193478, 6.52104739049924581790185829593, 7.43856900649409982691375733078, 8.315712647209521079979155237226
