| L(s) = 1 | − 1.31·3-s − 2.90·7-s − 1.28·9-s − 0.214·11-s − 13-s + 6.42·17-s − 2.21·19-s + 3.80·21-s − 4.68·23-s + 5.61·27-s + 8.70·29-s + 5.59·31-s + 0.280·33-s + 2.28·37-s + 1.31·39-s + 3.05·41-s − 6.36·43-s − 1.09·47-s + 1.42·49-s − 8.42·51-s − 6.23·53-s + 2.90·57-s + 9.26·59-s − 0.280·61-s + 3.71·63-s + 7.76·67-s + 6.14·69-s + ⋯ |
| L(s) = 1 | − 0.756·3-s − 1.09·7-s − 0.426·9-s − 0.0646·11-s − 0.277·13-s + 1.55·17-s − 0.507·19-s + 0.830·21-s − 0.977·23-s + 1.08·27-s + 1.61·29-s + 1.00·31-s + 0.0489·33-s + 0.374·37-s + 0.209·39-s + 0.476·41-s − 0.970·43-s − 0.159·47-s + 0.204·49-s − 1.18·51-s − 0.856·53-s + 0.384·57-s + 1.20·59-s − 0.0359·61-s + 0.468·63-s + 0.948·67-s + 0.740·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.028862837535083676417633438254, −6.82126401956356821762646584956, −6.42100800592798202217411094099, −5.74419057400454728161333764669, −5.08876333444008893535723820287, −4.15813411828844895999422831895, −3.20273345362771851962423778125, −2.57180967806542904623688944983, −1.06977528171549082018292440712, 0,
1.06977528171549082018292440712, 2.57180967806542904623688944983, 3.20273345362771851962423778125, 4.15813411828844895999422831895, 5.08876333444008893535723820287, 5.74419057400454728161333764669, 6.42100800592798202217411094099, 6.82126401956356821762646584956, 8.028862837535083676417633438254
