L(s) = 1 | + 5.48i·2-s − 3·3-s − 22.0·4-s + 13.3i·5-s − 16.4i·6-s − 21.4i·7-s − 77.3i·8-s + 9·9-s − 73.0·10-s + 19.0i·11-s + 66.2·12-s + 117.·14-s − 39.9i·15-s + 247.·16-s + 71.7·17-s + 49.3i·18-s + ⋯ |
L(s) = 1 | + 1.93i·2-s − 0.577·3-s − 2.76·4-s + 1.19i·5-s − 1.11i·6-s − 1.15i·7-s − 3.41i·8-s + 0.333·9-s − 2.31·10-s + 0.522i·11-s + 1.59·12-s + 2.24·14-s − 0.687i·15-s + 3.86·16-s + 1.02·17-s + 0.646i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.159493803\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.159493803\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 5.48iT - 8T^{2} \) |
| 5 | \( 1 - 13.3iT - 125T^{2} \) |
| 7 | \( 1 + 21.4iT - 343T^{2} \) |
| 11 | \( 1 - 19.0iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 71.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 37.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6.05iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 285. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 342. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 346. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 208. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 678. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 957. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 270. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 427. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 698. iT - 9.12e5T^{2} \) |
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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
−10.34206626592644696198489780506, −10.04815665747183870923356866434, −8.760297661921181078615125806338, −7.57994670340263861461883982562, −7.02831071550964234390197393141, −6.61106651824709906441712742026, −5.44113841894727597478811296815, −4.54054655362193761648761548751, −3.46232334113034538413910219789, −0.65455162804122289254578698461,
0.77444573066904663437004541807, 1.70972719723493968409670822122, 3.05643467674154914568082796495, 4.21094508914097386177945053823, 5.26631231483737970053747564263, 5.74361215592091794934236023787, 8.102298983014769660503846655590, 8.719997490443314627514719857117, 9.537921687166493295444302054214, 10.23300798234047392883589008659