| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 2.24·11-s − 5.41·13-s − 15-s − 4.41·17-s + 7.07·19-s − 21-s + 23-s + 25-s − 27-s + 1.24·29-s + 10.6·31-s + 2.24·33-s + 35-s − 3·37-s + 5.41·39-s − 1.58·41-s − 2·43-s + 45-s − 5.41·47-s − 6·49-s + 4.41·51-s − 6.41·53-s − 2.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.676·11-s − 1.50·13-s − 0.258·15-s − 1.07·17-s + 1.62·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.230·29-s + 1.91·31-s + 0.390·33-s + 0.169·35-s − 0.493·37-s + 0.866·39-s − 0.247·41-s − 0.304·43-s + 0.149·45-s − 0.789·47-s − 0.857·49-s + 0.618·51-s − 0.881·53-s − 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−7.74953599290933879962266125461, −6.99634598576095783817616187693, −6.46213577116685072095103128848, −5.40689523529663712690479257581, −5.01550542870153924814862024251, −4.41599738224735621597318883961, −3.06259212135660160400564661163, −2.38698678545669935520309868642, −1.29673460011563917580190857719, 0,
1.29673460011563917580190857719, 2.38698678545669935520309868642, 3.06259212135660160400564661163, 4.41599738224735621597318883961, 5.01550542870153924814862024251, 5.40689523529663712690479257581, 6.46213577116685072095103128848, 6.99634598576095783817616187693, 7.74953599290933879962266125461
