| L(s) = 1 | + 2.35·2-s + 3-s + 3.54·4-s − 0.993·5-s + 2.35·6-s + 3.62·8-s + 9-s − 2.33·10-s − 4.38·11-s + 3.54·12-s − 0.0172·13-s − 0.993·15-s + 1.45·16-s − 2.00·17-s + 2.35·18-s − 0.356·19-s − 3.51·20-s − 10.3·22-s − 6.85·23-s + 3.62·24-s − 4.01·25-s − 0.0405·26-s + 27-s − 3.21·29-s − 2.33·30-s − 5.04·31-s − 3.83·32-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.77·4-s − 0.444·5-s + 0.960·6-s + 1.28·8-s + 0.333·9-s − 0.739·10-s − 1.32·11-s + 1.02·12-s − 0.00478·13-s − 0.256·15-s + 0.362·16-s − 0.486·17-s + 0.554·18-s − 0.0816·19-s − 0.786·20-s − 2.20·22-s − 1.42·23-s + 0.739·24-s − 0.802·25-s − 0.00796·26-s + 0.192·27-s − 0.596·29-s − 0.427·30-s − 0.906·31-s − 0.677·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 + 0.993T + 5T^{2} \) |
| 11 | \( 1 + 4.38T + 11T^{2} \) |
| 13 | \( 1 + 0.0172T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 + 0.356T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 0.480T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 8.58T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.39087580935385689932310679232, −6.87151892447360952922202249333, −5.82068504308728520041384806462, −5.46156590301709871199324281701, −4.58964804107975787127088220102, −3.90301296975820463584064242653, −3.44267373996631731764332926496, −2.42578995501875840965903446102, −1.99320806577674556681471618244, 0,
1.99320806577674556681471618244, 2.42578995501875840965903446102, 3.44267373996631731764332926496, 3.90301296975820463584064242653, 4.58964804107975787127088220102, 5.46156590301709871199324281701, 5.82068504308728520041384806462, 6.87151892447360952922202249333, 7.39087580935385689932310679232
