L(s) = 1 | − 2.11·2-s + 2.49·4-s − 5-s − 4.83·7-s − 1.04·8-s + 2.11·10-s + 2.21·11-s − 2.26·13-s + 10.2·14-s − 2.77·16-s + 3.10·17-s − 5.93·19-s − 2.49·20-s − 4.69·22-s − 5.79·23-s + 25-s + 4.79·26-s − 12.0·28-s − 2.32·29-s + 4.47·31-s + 7.96·32-s − 6.59·34-s + 4.83·35-s − 8.23·37-s + 12.5·38-s + 1.04·40-s + 0.278·41-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 1.24·4-s − 0.447·5-s − 1.82·7-s − 0.369·8-s + 0.670·10-s + 0.668·11-s − 0.627·13-s + 2.73·14-s − 0.692·16-s + 0.754·17-s − 1.36·19-s − 0.557·20-s − 1.00·22-s − 1.20·23-s + 0.200·25-s + 0.940·26-s − 2.27·28-s − 0.432·29-s + 0.803·31-s + 1.40·32-s − 1.13·34-s + 0.816·35-s − 1.35·37-s + 2.04·38-s + 0.165·40-s + 0.0434·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2045254584\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2045254584\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 5.79T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 - 0.278T + 41T^{2} \) |
| 43 | \( 1 + 0.176T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 8.65T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 - 8.06T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 97 | \( 1 + 5.52T + 97T^{2} \) |
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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
−8.543780460767788644017514108180, −7.86570926182645227720502383751, −7.07191460419984150102116424347, −6.54854560279542335399131599390, −5.90561765764338513550873879229, −4.50506348867043270778594110557, −3.67030964179931905986957882141, −2.78843777062835554305536232888, −1.70020202082051440673147507162, −0.31640057740750242824242205185,
0.31640057740750242824242205185, 1.70020202082051440673147507162, 2.78843777062835554305536232888, 3.67030964179931905986957882141, 4.50506348867043270778594110557, 5.90561765764338513550873879229, 6.54854560279542335399131599390, 7.07191460419984150102116424347, 7.86570926182645227720502383751, 8.543780460767788644017514108180