| L(s) = 1 | + 2-s + 4-s + 1.76·5-s + 8-s + 1.76·10-s + 6.12·11-s + 0.760·13-s + 16-s + 6.84·17-s − 1.94·19-s + 1.76·20-s + 6.12·22-s − 0.421·23-s − 1.89·25-s + 0.760·26-s − 1.46·29-s + 7.70·31-s + 32-s + 6.84·34-s − 2.88·37-s − 1.94·38-s + 1.76·40-s + 6.94·41-s − 8.66·43-s + 6.12·44-s − 0.421·46-s + 1.66·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.787·5-s + 0.353·8-s + 0.556·10-s + 1.84·11-s + 0.211·13-s + 0.250·16-s + 1.65·17-s − 0.445·19-s + 0.393·20-s + 1.30·22-s − 0.0877·23-s − 0.379·25-s + 0.149·26-s − 0.271·29-s + 1.38·31-s + 0.176·32-s + 1.17·34-s − 0.474·37-s − 0.315·38-s + 0.278·40-s + 1.08·41-s − 1.32·43-s + 0.923·44-s − 0.0620·46-s + 0.242·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.944039029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.944039029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 - 0.760T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 - 1.98T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 0.306T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 3.63T + 97T^{2} \) |
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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
−7.77551959097459725484040511980, −6.86583229272002718500581567386, −6.33973043515108301056904068761, −5.82595567193704796624620922789, −5.11911490688705541065170348480, −4.17823050235400412435662472541, −3.65264642136979553333953062348, −2.79055364976688948434794313895, −1.73684441441446196309732902466, −1.11108933339772519787956192050,
1.11108933339772519787956192050, 1.73684441441446196309732902466, 2.79055364976688948434794313895, 3.65264642136979553333953062348, 4.17823050235400412435662472541, 5.11911490688705541065170348480, 5.82595567193704796624620922789, 6.33973043515108301056904068761, 6.86583229272002718500581567386, 7.77551959097459725484040511980
