| L(s) = 1 | − 3-s − 3.16·5-s + 4.57·7-s + 9-s + 2.47·11-s − 1.41·13-s + 3.16·15-s − 6.47·17-s − 2.47·19-s − 4.57·21-s − 5.65·23-s + 5.00·25-s − 27-s − 0.333·29-s + 10.2·31-s − 2.47·33-s − 14.4·35-s + 2.08·37-s + 1.41·39-s + 6.47·41-s − 10.4·43-s − 3.16·45-s + 13.9·49-s + 6.47·51-s − 5.32·53-s − 7.81·55-s + 2.47·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.41·5-s + 1.72·7-s + 0.333·9-s + 0.745·11-s − 0.392·13-s + 0.816·15-s − 1.56·17-s − 0.567·19-s − 0.998·21-s − 1.17·23-s + 1.00·25-s − 0.192·27-s − 0.0619·29-s + 1.83·31-s − 0.430·33-s − 2.44·35-s + 0.342·37-s + 0.226·39-s + 1.01·41-s − 1.59·43-s − 0.471·45-s + 1.99·49-s + 0.906·51-s − 0.731·53-s − 1.05·55-s + 0.327·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 0.333T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
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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
−8.320287358430937705719445442916, −7.66417869054201887976748859271, −6.90455805031291968159232503970, −6.13722399929469019077848912041, −4.96411715654378297002224495403, −4.35419493478129766048152168412, −4.04107433075932677937028421116, −2.45112814372945010978186534470, −1.36078778015542633915495682306, 0,
1.36078778015542633915495682306, 2.45112814372945010978186534470, 4.04107433075932677937028421116, 4.35419493478129766048152168412, 4.96411715654378297002224495403, 6.13722399929469019077848912041, 6.90455805031291968159232503970, 7.66417869054201887976748859271, 8.320287358430937705719445442916
