L(s) = 1 | + 0.839·3-s − 4.04·5-s − 2.98·7-s − 2.29·9-s + 4.04·11-s + 1.02·13-s − 3.39·15-s − 17-s − 1.21·19-s − 2.50·21-s − 5.22·23-s + 11.3·25-s − 4.44·27-s − 6.47·29-s − 9.80·31-s + 3.39·33-s + 12.0·35-s − 8.79·37-s + 0.861·39-s + 0.149·41-s + 9.75·43-s + 9.28·45-s + 12.1·47-s + 1.93·49-s − 0.839·51-s − 4.26·53-s − 16.3·55-s + ⋯ |
L(s) = 1 | + 0.484·3-s − 1.80·5-s − 1.12·7-s − 0.765·9-s + 1.22·11-s + 0.284·13-s − 0.876·15-s − 0.242·17-s − 0.279·19-s − 0.547·21-s − 1.08·23-s + 2.27·25-s − 0.855·27-s − 1.20·29-s − 1.76·31-s + 0.591·33-s + 2.04·35-s − 1.44·37-s + 0.137·39-s + 0.0233·41-s + 1.48·43-s + 1.38·45-s + 1.77·47-s + 0.276·49-s − 0.117·51-s − 0.585·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7082781886\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7082781886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 0.839T + 3T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 - 0.149T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 - 3.99T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 + 8.50T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.639830233791818701089941870638, −7.61221511460304657417936239900, −7.20899308139591001481209729470, −6.33461023231910367773769739869, −5.58498032456640013759887562387, −4.24926702074475806526721491284, −3.65082479055011087279830325099, −3.39743760394593268730855337662, −2.11036814545976095155363981533, −0.44654559219192470235044776733,
0.44654559219192470235044776733, 2.11036814545976095155363981533, 3.39743760394593268730855337662, 3.65082479055011087279830325099, 4.24926702074475806526721491284, 5.58498032456640013759887562387, 6.33461023231910367773769739869, 7.20899308139591001481209729470, 7.61221511460304657417936239900, 8.639830233791818701089941870638