| L(s) = 1 | − 2.14·3-s − 3.45·5-s − 3.94·7-s + 1.61·9-s − 11-s − 6.64·13-s + 7.42·15-s − 6.46·17-s + 3.77·19-s + 8.46·21-s + 5.69·23-s + 6.95·25-s + 2.98·27-s + 5.54·29-s − 2.25·31-s + 2.14·33-s + 13.6·35-s + 2.71·37-s + 14.2·39-s + 3.11·41-s − 6.42·43-s − 5.56·45-s − 2.79·47-s + 8.53·49-s + 13.8·51-s + 6.48·53-s + 3.45·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s − 1.54·5-s − 1.48·7-s + 0.536·9-s − 0.301·11-s − 1.84·13-s + 1.91·15-s − 1.56·17-s + 0.864·19-s + 1.84·21-s + 1.18·23-s + 1.39·25-s + 0.574·27-s + 1.03·29-s − 0.404·31-s + 0.373·33-s + 2.30·35-s + 0.446·37-s + 2.28·39-s + 0.486·41-s − 0.979·43-s − 0.830·45-s − 0.407·47-s + 1.21·49-s + 1.94·51-s + 0.890·53-s + 0.466·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 + 2.79T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 3.75T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 0.436T + 97T^{2} \) |
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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
−7.45575683938792933394817493788, −6.85530314670004093611017642083, −6.60453829847689335798578617916, −5.45986685648347947418179974975, −4.83853766301517012732177868740, −4.24588403861654712273775650719, −3.19350995447949085982590845656, −2.61352133212846584038589867544, −0.63290861732345964362602620422, 0,
0.63290861732345964362602620422, 2.61352133212846584038589867544, 3.19350995447949085982590845656, 4.24588403861654712273775650719, 4.83853766301517012732177868740, 5.45986685648347947418179974975, 6.60453829847689335798578617916, 6.85530314670004093611017642083, 7.45575683938792933394817493788
