| L(s) = 1 | + 2.73·3-s − 5-s − 3.46·7-s + 4.46·9-s + 4.73·11-s + 13-s − 2.73·15-s + 3.46·17-s + 3.26·19-s − 9.46·21-s + 8.19·23-s + 25-s + 3.99·27-s − 5.46·29-s + 4.73·31-s + 12.9·33-s + 3.46·35-s − 2.92·37-s + 2.73·39-s − 11.4·41-s + 2.73·43-s − 4.46·45-s + 11.4·47-s + 4.99·49-s + 9.46·51-s + 11.4·53-s − 4.73·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.447·5-s − 1.30·7-s + 1.48·9-s + 1.42·11-s + 0.277·13-s − 0.705·15-s + 0.840·17-s + 0.749·19-s − 2.06·21-s + 1.70·23-s + 0.200·25-s + 0.769·27-s − 1.01·29-s + 0.849·31-s + 2.25·33-s + 0.585·35-s − 0.481·37-s + 0.437·39-s − 1.79·41-s + 0.416·43-s − 0.665·45-s + 1.67·47-s + 0.714·49-s + 1.32·51-s + 1.57·53-s − 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.541634706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.541634706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.607289652788812170525813313921, −9.061387198178810891905373000907, −8.533282882079795593937326184557, −7.31366823778085007945657570058, −6.95971742468071165864179048534, −5.73974034555109740290873394545, −4.20298920860596348802360263558, −3.40981136754528865536639413729, −2.93705634094810328189287060876, −1.29342481061562879556021312902,
1.29342481061562879556021312902, 2.93705634094810328189287060876, 3.40981136754528865536639413729, 4.20298920860596348802360263558, 5.73974034555109740290873394545, 6.95971742468071165864179048534, 7.31366823778085007945657570058, 8.533282882079795593937326184557, 9.061387198178810891905373000907, 9.607289652788812170525813313921
