| L(s) = 1 | − 1.37·2-s − 6.11·4-s + 5.11·7-s + 19.3·8-s − 55.9·11-s − 37.5·13-s − 7.02·14-s + 22.3·16-s + 23.6·17-s + 39.0·19-s + 76.8·22-s + 71.0·23-s + 51.5·26-s − 31.2·28-s + 28.3·29-s + 12.8·31-s − 185.·32-s − 32.4·34-s + 180.·37-s − 53.5·38-s + 215.·41-s − 61.2·43-s + 342.·44-s − 97.5·46-s − 61.8·47-s − 316.·49-s + 229.·52-s + ⋯ |
| L(s) = 1 | − 0.485·2-s − 0.764·4-s + 0.276·7-s + 0.856·8-s − 1.53·11-s − 0.801·13-s − 0.134·14-s + 0.349·16-s + 0.337·17-s + 0.471·19-s + 0.744·22-s + 0.644·23-s + 0.389·26-s − 0.211·28-s + 0.181·29-s + 0.0746·31-s − 1.02·32-s − 0.163·34-s + 0.800·37-s − 0.228·38-s + 0.820·41-s − 0.217·43-s + 1.17·44-s − 0.312·46-s − 0.192·47-s − 0.923·49-s + 0.613·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.37T + 8T^{2} \) |
| 7 | \( 1 - 5.11T + 343T^{2} \) |
| 11 | \( 1 + 55.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 71.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 28.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 61.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 61.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 270.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 713.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 404.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 75.0T + 9.12e5T^{2} \) |
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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
−8.329473670039198788071047423341, −7.75254506046487691132432071673, −7.20200057680276313778535889193, −5.83734158005811905783195171427, −5.05221691894350691474010169319, −4.55762776552464795565990851977, −3.29056720081578573632076604100, −2.32034612835706651814651496064, −0.999741528731994651366378859782, 0,
0.999741528731994651366378859782, 2.32034612835706651814651496064, 3.29056720081578573632076604100, 4.55762776552464795565990851977, 5.05221691894350691474010169319, 5.83734158005811905783195171427, 7.20200057680276313778535889193, 7.75254506046487691132432071673, 8.329473670039198788071047423341
