| L(s) = 1 | − 1.91·3-s + 0.891·5-s + 4.37·7-s + 0.681·9-s − 0.506·11-s − 2.73·13-s − 1.70·15-s − 17-s − 4.67·19-s − 8.39·21-s − 5.20·23-s − 4.20·25-s + 4.44·27-s + 7.35·29-s − 5.22·31-s + 0.971·33-s + 3.89·35-s − 5.81·37-s + 5.23·39-s − 5.15·41-s + 0.273·43-s + 0.607·45-s − 1.21·47-s + 12.1·49-s + 1.91·51-s + 1.84·53-s − 0.451·55-s + ⋯ |
| L(s) = 1 | − 1.10·3-s + 0.398·5-s + 1.65·7-s + 0.227·9-s − 0.152·11-s − 0.757·13-s − 0.441·15-s − 0.242·17-s − 1.07·19-s − 1.83·21-s − 1.08·23-s − 0.841·25-s + 0.856·27-s + 1.36·29-s − 0.938·31-s + 0.169·33-s + 0.658·35-s − 0.956·37-s + 0.838·39-s − 0.804·41-s + 0.0417·43-s + 0.0905·45-s − 0.176·47-s + 1.73·49-s + 0.268·51-s + 0.254·53-s − 0.0608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255935201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255935201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 - 0.891T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 + 0.506T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 0.273T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−7.941785334452863882794005446030, −7.01023144971953199626766473043, −6.36410925949304471906998619598, −5.64793969069391441942628875950, −5.03009028279033763350366927871, −4.65238489874393326502666597828, −3.72218158177898120945860521264, −2.26223594131219106310240923041, −1.86352405789169346860332812493, −0.57659807507609177733836096868,
0.57659807507609177733836096868, 1.86352405789169346860332812493, 2.26223594131219106310240923041, 3.72218158177898120945860521264, 4.65238489874393326502666597828, 5.03009028279033763350366927871, 5.64793969069391441942628875950, 6.36410925949304471906998619598, 7.01023144971953199626766473043, 7.941785334452863882794005446030
