| L(s) = 1 | + 3-s − 3.64·5-s + 2.86·7-s + 9-s − 3.28·11-s − 4.30·13-s − 3.64·15-s + 6.11·17-s − 0.135·19-s + 2.86·21-s + 2.55·23-s + 8.25·25-s + 27-s + 2.37·29-s − 10.0·31-s − 3.28·33-s − 10.4·35-s + 3.35·37-s − 4.30·39-s + 9.70·41-s − 10.1·43-s − 3.64·45-s + 4.02·47-s + 1.19·49-s + 6.11·51-s + 1.81·53-s + 11.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.62·5-s + 1.08·7-s + 0.333·9-s − 0.991·11-s − 1.19·13-s − 0.939·15-s + 1.48·17-s − 0.0309·19-s + 0.624·21-s + 0.533·23-s + 1.65·25-s + 0.192·27-s + 0.440·29-s − 1.81·31-s − 0.572·33-s − 1.76·35-s + 0.551·37-s − 0.689·39-s + 1.51·41-s − 1.55·43-s − 0.542·45-s + 0.586·47-s + 0.170·49-s + 0.855·51-s + 0.249·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631460957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631460957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.135T + 19T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 2.37T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 - 1.81T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 1.10T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 2.19T + 97T^{2} \) |
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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
−7.85717529326029786575896336129, −7.54599087501736605451262127118, −7.23133394833540648076295792110, −5.77557440232983412831874475034, −4.89766369183869231586501356033, −4.56204678100309886403660860260, −3.53471528408457786255398256895, −2.96274144309492416822955112460, −1.92302220932472346291672356341, −0.64459905179545385560231916846,
0.64459905179545385560231916846, 1.92302220932472346291672356341, 2.96274144309492416822955112460, 3.53471528408457786255398256895, 4.56204678100309886403660860260, 4.89766369183869231586501356033, 5.77557440232983412831874475034, 7.23133394833540648076295792110, 7.54599087501736605451262127118, 7.85717529326029786575896336129
