| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s + 2·11-s + 2·12-s − 13-s + 14-s + 16-s + 3·17-s − 18-s − 2·19-s − 2·21-s − 2·22-s − 7·23-s − 2·24-s + 26-s − 4·27-s − 28-s − 5·29-s − 10·31-s − 32-s + 4·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s − 0.436·21-s − 0.426·22-s − 1.45·23-s − 0.408·24-s + 0.196·26-s − 0.769·27-s − 0.188·28-s − 0.928·29-s − 1.79·31-s − 0.176·32-s + 0.696·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155103714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155103714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.82060681498953, −13.08654901433465, −13.03557466596810, −12.01015887639776, −11.93577203406124, −11.26647298872176, −10.61875623463631, −10.02660811334764, −9.768009446140142, −9.224132150677376, −8.690253117277000, −8.465523605317950, −7.733030595103523, −7.427559652968764, −6.859349702932524, −6.240901377600809, −5.634811909247760, −5.159294023703343, −4.035924679993714, −3.719952423547024, −3.290439959761230, −2.449464496630084, −1.980023308632048, −1.473831061876617, −0.3289195290643153,
0.3289195290643153, 1.473831061876617, 1.980023308632048, 2.449464496630084, 3.290439959761230, 3.719952423547024, 4.035924679993714, 5.159294023703343, 5.634811909247760, 6.240901377600809, 6.859349702932524, 7.427559652968764, 7.733030595103523, 8.465523605317950, 8.690253117277000, 9.224132150677376, 9.768009446140142, 10.02660811334764, 10.61875623463631, 11.26647298872176, 11.93577203406124, 12.01015887639776, 13.03557466596810, 13.08654901433465, 13.82060681498953
