| L(s) = 1 | + 4i·2-s + 15.0i·3-s − 16·4-s − 60.1·6-s − 52.9i·7-s − 64i·8-s + 16.7·9-s − 259.·11-s − 240. i·12-s − 169i·13-s + 211.·14-s + 256·16-s + 2.27e3i·17-s + 67.0i·18-s − 730.·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.964i·3-s − 0.5·4-s − 0.682·6-s − 0.408i·7-s − 0.353i·8-s + 0.0689·9-s − 0.646·11-s − 0.482i·12-s − 0.277i·13-s + 0.288·14-s + 0.250·16-s + 1.90i·17-s + 0.0487i·18-s − 0.464·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.04579176843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04579176843\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 3 | \( 1 - 15.0iT - 243T^{2} \) |
| 7 | \( 1 + 52.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 259.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.27e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 730.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.97e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 949.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 225.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 954. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.73e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.00e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 6.06e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.61e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 326.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.35e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.32e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.01e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5iT - 8.58e9T^{2} \) |
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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
−10.40790619474296409505991376080, −9.706302855375072421957104665301, −8.662661688000375134470076984949, −7.961142929550164673922982855255, −6.96971442961672496141961318981, −5.94114598729131222646953846956, −5.05001026825667721923644286311, −4.14097785526694921095865471638, −3.38987187551664775892646897918, −1.62731563622905469151359677868,
0.01065823892917375869584017051, 1.03981652324935842161681350023, 2.20753833550490666911552179309, 2.89563185852739635515068884222, 4.39951497146088788881570406557, 5.29569158155226873315245948461, 6.50986139136233725694594345556, 7.31812029249037165527143255682, 8.212509256201341142292450576972, 9.110418212752716757991953245728
