| L(s) = 1 | + 1.36·2-s − 0.141·4-s − 3.14·5-s − 7-s − 2.91·8-s − 4.28·10-s − 4.77·13-s − 1.36·14-s − 3.69·16-s + 4.77·17-s − 7.00·19-s + 0.443·20-s − 5.14·23-s + 4.86·25-s − 6.51·26-s + 0.141·28-s − 7.00·29-s − 3.63·31-s + 0.797·32-s + 6.51·34-s + 3.14·35-s − 9.86·37-s − 9.55·38-s + 9.17·40-s + 3.22·41-s + 4.28·43-s − 7.00·46-s + ⋯ |
| L(s) = 1 | + 0.964·2-s − 0.0706·4-s − 1.40·5-s − 0.377·7-s − 1.03·8-s − 1.35·10-s − 1.32·13-s − 0.364·14-s − 0.924·16-s + 1.15·17-s − 1.60·19-s + 0.0992·20-s − 1.07·23-s + 0.973·25-s − 1.27·26-s + 0.0267·28-s − 1.30·29-s − 0.653·31-s + 0.141·32-s + 1.11·34-s + 0.530·35-s − 1.62·37-s − 1.55·38-s + 1.45·40-s + 0.503·41-s + 0.653·43-s − 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4393032680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4393032680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 13 | \( 1 + 4.77T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 - 0.778T + 47T^{2} \) |
| 53 | \( 1 + 2.28T + 53T^{2} \) |
| 59 | \( 1 - 0.363T + 59T^{2} \) |
| 61 | \( 1 - 3.22T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 + 1.55T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−7.75313156583162807215112324886, −7.21934426965194010658859247763, −6.39598709050696225160017916355, −5.58143082535354746540985385948, −4.99030109203501805800038142300, −4.04758106767704680895208379525, −3.87916491334639476478821934364, −3.02747000791118471317600951172, −2.07862807394325359527741541216, −0.26729995558272639143013498662,
0.26729995558272639143013498662, 2.07862807394325359527741541216, 3.02747000791118471317600951172, 3.87916491334639476478821934364, 4.04758106767704680895208379525, 4.99030109203501805800038142300, 5.58143082535354746540985385948, 6.39598709050696225160017916355, 7.21934426965194010658859247763, 7.75313156583162807215112324886
