| L(s) = 1 | − 2.12·3-s + 1.48·5-s + 1.51·9-s + 3.15·11-s + 6.76·13-s − 3.15·15-s + 2·17-s − 1.03·19-s + 4.24·23-s − 2.79·25-s + 3.15·27-s + 29-s − 1.87·31-s − 6.70·33-s − 0.969·37-s − 14.3·39-s − 7.52·41-s + 1.09·43-s + 2.24·45-s − 9.34·47-s − 7·49-s − 4.24·51-s + 5.73·53-s + 4.68·55-s + 2.18·57-s + 8.24·59-s − 10.4·61-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 0.664·5-s + 0.505·9-s + 0.951·11-s + 1.87·13-s − 0.814·15-s + 0.485·17-s − 0.236·19-s + 0.886·23-s − 0.559·25-s + 0.607·27-s + 0.185·29-s − 0.336·31-s − 1.16·33-s − 0.159·37-s − 2.30·39-s − 1.17·41-s + 0.166·43-s + 0.335·45-s − 1.36·47-s − 49-s − 0.595·51-s + 0.787·53-s + 0.631·55-s + 0.289·57-s + 1.07·59-s − 1.34·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007093070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007093070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 0.969T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 9.34T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.96T + 97T^{2} \) |
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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
−11.97466315845776966881431428721, −11.25254548709380443504574172207, −10.51090972982363681890442212292, −9.380260212434287112257553064602, −8.362053558161570241649850447912, −6.66957772962695312014416554727, −6.11170769400516832432523476697, −5.13586615803755969887543163989, −3.62959767998240398835010148190, −1.34872573937947201200284517279,
1.34872573937947201200284517279, 3.62959767998240398835010148190, 5.13586615803755969887543163989, 6.11170769400516832432523476697, 6.66957772962695312014416554727, 8.362053558161570241649850447912, 9.380260212434287112257553064602, 10.51090972982363681890442212292, 11.25254548709380443504574172207, 11.97466315845776966881431428721
