| L(s) = 1 | + 3.39·5-s − 0.822·7-s − 4.90·13-s − 4.90·17-s + 7.61·19-s − 2.97·23-s + 6.53·25-s − 5.34·29-s − 4.39·31-s − 2.79·35-s + 5.97·37-s + 2.55·41-s − 8.43·43-s − 11.7·47-s − 6.32·49-s + 12.3·53-s + 6.97·59-s − 1.67·61-s − 16.6·65-s − 2.57·67-s + 9.10·71-s − 4.68·73-s − 8.68·79-s − 5.64·83-s − 16.6·85-s − 9.21·89-s + 4.03·91-s + ⋯ |
| L(s) = 1 | + 1.51·5-s − 0.310·7-s − 1.36·13-s − 1.19·17-s + 1.74·19-s − 0.619·23-s + 1.30·25-s − 0.992·29-s − 0.789·31-s − 0.471·35-s + 0.981·37-s + 0.398·41-s − 1.28·43-s − 1.70·47-s − 0.903·49-s + 1.69·53-s + 0.907·59-s − 0.213·61-s − 2.06·65-s − 0.314·67-s + 1.08·71-s − 0.548·73-s − 0.977·79-s − 0.619·83-s − 1.80·85-s − 0.976·89-s + 0.423·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 0.822T + 7T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 + 1.67T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.27815841738138161319816319030, −6.73525887833179631299191757600, −6.00350926288472847790772954011, −5.33321143488476506446610935408, −4.89821491785922804405980138930, −3.82830793427409532231571256739, −2.83674504244152182880282050959, −2.22320569172799296505457693688, −1.45173327136704299946162696444, 0,
1.45173327136704299946162696444, 2.22320569172799296505457693688, 2.83674504244152182880282050959, 3.82830793427409532231571256739, 4.89821491785922804405980138930, 5.33321143488476506446610935408, 6.00350926288472847790772954011, 6.73525887833179631299191757600, 7.27815841738138161319816319030
