| L(s) = 1 | + 1.73·3-s + 0.367·5-s − 0.998·7-s − 0.00381·9-s + 2.58·11-s + 3.65·13-s + 0.636·15-s + 1.48·17-s − 5.32·19-s − 1.72·21-s − 7.18·23-s − 4.86·25-s − 5.19·27-s + 2.32·29-s − 8.43·31-s + 4.47·33-s − 0.367·35-s + 7.73·37-s + 6.31·39-s − 8.54·41-s + 2.60·43-s − 0.00140·45-s − 3.71·47-s − 6.00·49-s + 2.57·51-s − 7.53·53-s + 0.951·55-s + ⋯ |
| L(s) = 1 | + 0.999·3-s + 0.164·5-s − 0.377·7-s − 0.00127·9-s + 0.780·11-s + 1.01·13-s + 0.164·15-s + 0.361·17-s − 1.22·19-s − 0.377·21-s − 1.49·23-s − 0.972·25-s − 1.00·27-s + 0.431·29-s − 1.51·31-s + 0.779·33-s − 0.0620·35-s + 1.27·37-s + 1.01·39-s − 1.33·41-s + 0.397·43-s − 0.000209·45-s − 0.541·47-s − 0.857·49-s + 0.360·51-s − 1.03·53-s + 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 0.367T + 5T^{2} \) |
| 7 | \( 1 + 0.998T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 3.65T + 13T^{2} \) |
| 17 | \( 1 - 1.48T + 17T^{2} \) |
| 19 | \( 1 + 5.32T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 2.60T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 7.75T + 71T^{2} \) |
| 73 | \( 1 + 6.59T + 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 2.84T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 7.84T + 97T^{2} \) |
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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
−7.88257983249550408721926885922, −6.63889570863551013278578027536, −6.22423393206908490259945006597, −5.57778960796175206857792023189, −4.36319128955533697871718157728, −3.74662033972334558150830730841, −3.24328011340126864683505143917, −2.16771778064613623382148231071, −1.58117572648381164171540774435, 0,
1.58117572648381164171540774435, 2.16771778064613623382148231071, 3.24328011340126864683505143917, 3.74662033972334558150830730841, 4.36319128955533697871718157728, 5.57778960796175206857792023189, 6.22423393206908490259945006597, 6.63889570863551013278578027536, 7.88257983249550408721926885922
