L(s) = 1 | − 0.766·3-s + 5-s + 7-s − 2.41·9-s − 11-s − 4.45·13-s − 0.766·15-s + 2.76·17-s − 7.21·19-s − 0.766·21-s − 7.09·23-s + 25-s + 4.14·27-s + 2·29-s − 4.64·31-s + 0.766·33-s + 35-s + 11.0·37-s + 3.41·39-s + 8.17·41-s − 10.6·43-s − 2.41·45-s + 4.91·47-s + 49-s − 2.12·51-s − 10.1·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.442·3-s + 0.447·5-s + 0.377·7-s − 0.804·9-s − 0.301·11-s − 1.23·13-s − 0.197·15-s + 0.671·17-s − 1.65·19-s − 0.167·21-s − 1.47·23-s + 0.200·25-s + 0.798·27-s + 0.371·29-s − 0.834·31-s + 0.133·33-s + 0.169·35-s + 1.82·37-s + 0.546·39-s + 1.27·41-s − 1.62·43-s − 0.359·45-s + 0.717·47-s + 0.142·49-s − 0.297·51-s − 1.39·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.153622579\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.153622579\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 0.766T + 3T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 4.91T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 - 8.57T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.14T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
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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
−8.073940728185639536197313584709, −7.43969666210898119778292681778, −6.37911686174454200404587309002, −6.02513531183015769612801788048, −5.18037586049611515335963419902, −4.64481562519406027074480679096, −3.67890115305164753701206912898, −2.50993271048263645515647090420, −2.05576996891357544651854160462, −0.54498323475716917015383401219,
0.54498323475716917015383401219, 2.05576996891357544651854160462, 2.50993271048263645515647090420, 3.67890115305164753701206912898, 4.64481562519406027074480679096, 5.18037586049611515335963419902, 6.02513531183015769612801788048, 6.37911686174454200404587309002, 7.43969666210898119778292681778, 8.073940728185639536197313584709