| L(s) = 1 | − 2.44·3-s − 2.61·5-s + 0.969·7-s + 2.99·9-s − 0.473·11-s − 6.64·13-s + 6.40·15-s + 3.15·17-s − 7.48·19-s − 2.37·21-s − 7.33·23-s + 1.84·25-s + 0.0183·27-s − 0.238·29-s + 0.258·31-s + 1.16·33-s − 2.53·35-s − 5.02·37-s + 16.2·39-s − 6.95·41-s + 9.54·43-s − 7.83·45-s − 47-s − 6.05·49-s − 7.73·51-s − 11.0·53-s + 1.24·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 1.17·5-s + 0.366·7-s + 0.997·9-s − 0.142·11-s − 1.84·13-s + 1.65·15-s + 0.766·17-s − 1.71·19-s − 0.518·21-s − 1.52·23-s + 0.369·25-s + 0.00353·27-s − 0.0443·29-s + 0.0463·31-s + 0.201·33-s − 0.429·35-s − 0.825·37-s + 2.60·39-s − 1.08·41-s + 1.45·43-s − 1.16·45-s − 0.145·47-s − 0.865·49-s − 1.08·51-s − 1.51·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01514411522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01514411522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 - 0.969T + 7T^{2} \) |
| 11 | \( 1 + 0.473T + 11T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 + 0.238T + 29T^{2} \) |
| 31 | \( 1 - 0.258T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 0.263T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.996371941513995367624576951718, −7.30423356125019009311877243198, −6.69584164964884273440403970401, −5.86408136742774638604962844492, −5.18977958547258321441388224009, −4.48924951637295433686801619106, −4.01899583921070937425903985666, −2.77583814945478363722173029474, −1.67314038457574259980441520089, −0.06763268975986686654492930509,
0.06763268975986686654492930509, 1.67314038457574259980441520089, 2.77583814945478363722173029474, 4.01899583921070937425903985666, 4.48924951637295433686801619106, 5.18977958547258321441388224009, 5.86408136742774638604962844492, 6.69584164964884273440403970401, 7.30423356125019009311877243198, 7.996371941513995367624576951718
