| L(s) = 1 | − 0.879·2-s − 1.22·4-s + 2.18·7-s + 2.83·8-s − 0.162·11-s − 2.41·13-s − 1.92·14-s − 0.0418·16-s + 3·17-s + 3.59·19-s + 0.142·22-s + 2.83·23-s + 2.12·26-s − 2.68·28-s − 6.71·29-s − 5.18·31-s − 5.63·32-s − 2.63·34-s + 6.63·37-s − 3.16·38-s + 5.80·41-s + 6.22·43-s + 0.199·44-s − 2.49·46-s + 7.39·47-s − 2.22·49-s + 2.95·52-s + ⋯ |
| L(s) = 1 | − 0.621·2-s − 0.613·4-s + 0.825·7-s + 1.00·8-s − 0.0489·11-s − 0.668·13-s − 0.513·14-s − 0.0104·16-s + 0.727·17-s + 0.825·19-s + 0.0304·22-s + 0.591·23-s + 0.415·26-s − 0.506·28-s − 1.24·29-s − 0.931·31-s − 0.996·32-s − 0.452·34-s + 1.09·37-s − 0.513·38-s + 0.905·41-s + 0.949·43-s + 0.0300·44-s − 0.367·46-s + 1.07·47-s − 0.318·49-s + 0.410·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292054834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292054834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 0.162T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 1.26T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 - 3.90T + 97T^{2} \) |
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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
−7.915475889167526799838515474155, −7.64901318129274205588914760309, −7.00344915286461969251024106589, −5.62776032283456124287936366983, −5.32664882251355969332092132545, −4.44008021993274159914466515105, −3.77625435613680924832916288952, −2.64580802490679142630004577841, −1.58726570522028227413484264095, −0.70361751561108712567944034427,
0.70361751561108712567944034427, 1.58726570522028227413484264095, 2.64580802490679142630004577841, 3.77625435613680924832916288952, 4.44008021993274159914466515105, 5.32664882251355969332092132545, 5.62776032283456124287936366983, 7.00344915286461969251024106589, 7.64901318129274205588914760309, 7.915475889167526799838515474155
