| L(s) = 1 | + 0.551·3-s − 2.33·5-s + 1.71·7-s − 2.69·9-s − 3.53·11-s + 4.69·13-s − 1.28·15-s + 5.32·17-s + 19-s + 0.945·21-s − 3.91·23-s + 0.440·25-s − 3.14·27-s − 0.324·29-s + 3.84·31-s − 1.95·33-s − 3.99·35-s − 2.10·37-s + 2.59·39-s − 11.2·41-s + 4.62·43-s + 6.28·45-s + 2.89·47-s − 4.06·49-s + 2.93·51-s − 0.546·53-s + 8.24·55-s + ⋯ |
| L(s) = 1 | + 0.318·3-s − 1.04·5-s + 0.647·7-s − 0.898·9-s − 1.06·11-s + 1.30·13-s − 0.332·15-s + 1.29·17-s + 0.229·19-s + 0.206·21-s − 0.817·23-s + 0.0880·25-s − 0.604·27-s − 0.0603·29-s + 0.690·31-s − 0.339·33-s − 0.675·35-s − 0.345·37-s + 0.414·39-s − 1.76·41-s + 0.704·43-s + 0.937·45-s + 0.421·47-s − 0.580·49-s + 0.411·51-s − 0.0750·53-s + 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 0.551T + 3T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + 0.324T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 0.546T + 53T^{2} \) |
| 59 | \( 1 - 0.300T + 59T^{2} \) |
| 61 | \( 1 - 9.09T + 61T^{2} \) |
| 67 | \( 1 - 2.81T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 + 0.382T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−7.918932718787266530937825428864, −7.32326575596702770075972878192, −6.19375892566273243034437756398, −5.54483324590928608289375752518, −4.87405785494463579373072237789, −3.81107546624546022833385578266, −3.38207075722045803760135891281, −2.45596973089389271541625515594, −1.26960611677553576538793974657, 0,
1.26960611677553576538793974657, 2.45596973089389271541625515594, 3.38207075722045803760135891281, 3.81107546624546022833385578266, 4.87405785494463579373072237789, 5.54483324590928608289375752518, 6.19375892566273243034437756398, 7.32326575596702770075972878192, 7.918932718787266530937825428864
