| L(s) = 1 | + 0.576·2-s + 1.89·3-s − 1.66·4-s + 1.09·6-s − 4.24·7-s − 2.11·8-s + 0.576·9-s + 1.09·11-s − 3.15·12-s − 4.69·13-s − 2.44·14-s + 2.11·16-s − 2.80·17-s + 0.332·18-s − 4.20·19-s − 8.02·21-s + 0.629·22-s + 4.24·23-s − 4·24-s − 2.70·26-s − 4.58·27-s + 7.07·28-s + 6.02·29-s − 31-s + 5.44·32-s + 2.06·33-s − 1.61·34-s + ⋯ |
| L(s) = 1 | + 0.407·2-s + 1.09·3-s − 0.833·4-s + 0.445·6-s − 1.60·7-s − 0.747·8-s + 0.192·9-s + 0.328·11-s − 0.910·12-s − 1.30·13-s − 0.654·14-s + 0.528·16-s − 0.679·17-s + 0.0783·18-s − 0.964·19-s − 1.75·21-s + 0.134·22-s + 0.884·23-s − 0.816·24-s − 0.530·26-s − 0.881·27-s + 1.33·28-s + 1.11·29-s − 0.179·31-s + 0.963·32-s + 0.359·33-s − 0.276·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.576T + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 + 4.20T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 0.716T + 59T^{2} \) |
| 61 | \( 1 - 9.78T + 61T^{2} \) |
| 67 | \( 1 + 3.27T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 2.50T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 - 4.55T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−9.777385855320466212772479483293, −8.848843700837542322562761475022, −8.638118403667585901979501881813, −7.20033706077984408526017544244, −6.45230251600721391421036475253, −5.25322988507831857249031619534, −4.16058849642353679289807950630, −3.28448304672870654543280156432, −2.51009516057417714292644617883, 0,
2.51009516057417714292644617883, 3.28448304672870654543280156432, 4.16058849642353679289807950630, 5.25322988507831857249031619534, 6.45230251600721391421036475253, 7.20033706077984408526017544244, 8.638118403667585901979501881813, 8.848843700837542322562761475022, 9.777385855320466212772479483293
