| L(s) = 1 | + 5-s − 7-s − 1.30·11-s + 3.60·13-s − 5.30·17-s + 19-s + 7.60·23-s − 4·25-s + 0.697·29-s − 4.69·31-s − 35-s − 3.60·37-s + 3.30·41-s − 7.21·43-s + 0.394·47-s + 49-s − 6.90·53-s − 1.30·55-s + 1.60·59-s − 6.21·61-s + 3.60·65-s − 0.0916·67-s + 12.8·71-s + 4.90·73-s + 1.30·77-s − 14·79-s − 7.51·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.392·11-s + 1.00·13-s − 1.28·17-s + 0.229·19-s + 1.58·23-s − 0.800·25-s + 0.129·29-s − 0.843·31-s − 0.169·35-s − 0.592·37-s + 0.515·41-s − 1.09·43-s + 0.0575·47-s + 0.142·49-s − 0.948·53-s − 0.175·55-s + 0.209·59-s − 0.795·61-s + 0.447·65-s − 0.0111·67-s + 1.52·71-s + 0.574·73-s + 0.148·77-s − 1.57·79-s − 0.824·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 - 0.697T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 0.394T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 + 0.0916T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−7.19054524223103678107597163486, −6.68324536357205240430201925708, −6.01296835659468474592498428011, −5.34133344286343828268205149554, −4.62784896402695750351576166627, −3.73922481929958398787677790064, −3.04905394863737767405049431209, −2.16652981285852268534514010607, −1.28931383978205027998475117348, 0,
1.28931383978205027998475117348, 2.16652981285852268534514010607, 3.04905394863737767405049431209, 3.73922481929958398787677790064, 4.62784896402695750351576166627, 5.34133344286343828268205149554, 6.01296835659468474592498428011, 6.68324536357205240430201925708, 7.19054524223103678107597163486
