| L(s) = 1 | + 1.64·5-s + 1.01·7-s − 5.21·11-s + 2.55·13-s − 3.56·17-s − 3.09·19-s − 23-s − 2.29·25-s − 1.44·29-s − 1.44·31-s + 1.66·35-s + 6.10·37-s − 4.73·41-s − 11.3·43-s + 0.0221·47-s − 5.97·49-s − 7.66·53-s − 8.57·55-s − 3.28·59-s + 7.21·61-s + 4.20·65-s − 11.4·67-s + 0.736·71-s + 11.6·73-s − 5.26·77-s + 2.12·79-s − 9.21·83-s + ⋯ |
| L(s) = 1 | + 0.736·5-s + 0.382·7-s − 1.57·11-s + 0.708·13-s − 0.864·17-s − 0.709·19-s − 0.208·23-s − 0.458·25-s − 0.268·29-s − 0.259·31-s + 0.281·35-s + 1.00·37-s − 0.739·41-s − 1.72·43-s + 0.00323·47-s − 0.853·49-s − 1.05·53-s − 1.15·55-s − 0.427·59-s + 0.923·61-s + 0.521·65-s − 1.40·67-s + 0.0874·71-s + 1.36·73-s − 0.600·77-s + 0.238·79-s − 1.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.0221T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 + 3.28T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 0.736T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.12T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−8.238739202761943357495012904181, −7.68455015198018126430601996264, −6.61440618000315278796213407726, −6.01662226450021186253691111378, −5.19989058585897863511542248374, −4.55376041138251611768916362933, −3.42869974558709642703049158413, −2.39008684691880950867858469667, −1.68963621623144903414850946665, 0,
1.68963621623144903414850946665, 2.39008684691880950867858469667, 3.42869974558709642703049158413, 4.55376041138251611768916362933, 5.19989058585897863511542248374, 6.01662226450021186253691111378, 6.61440618000315278796213407726, 7.68455015198018126430601996264, 8.238739202761943357495012904181
