| L(s) = 1 | − i·2-s + 3.14i·3-s − 4-s + 3.14·6-s + 3.89i·7-s + i·8-s − 6.89·9-s − 4.29·11-s − 3.14i·12-s + 4.34i·13-s + 3.89·14-s + 16-s − 1.60i·17-s + 6.89i·18-s + 1.20·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.81i·3-s − 0.5·4-s + 1.28·6-s + 1.47i·7-s + 0.353i·8-s − 2.29·9-s − 1.29·11-s − 0.907i·12-s + 1.20i·13-s + 1.04·14-s + 0.250·16-s − 0.388i·17-s + 1.62i·18-s + 0.275·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6439348564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6439348564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.14iT - 3T^{2} \) |
| 7 | \( 1 - 3.89iT - 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 4.34iT - 13T^{2} \) |
| 17 | \( 1 + 1.60iT - 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 8.34iT - 23T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 9.78iT - 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 3.60iT - 43T^{2} \) |
| 47 | \( 1 + 8.58iT - 47T^{2} \) |
| 53 | \( 1 - 4.80iT - 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 9.08iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2.39iT - 73T^{2} \) |
| 79 | \( 1 + 7.55T + 79T^{2} \) |
| 83 | \( 1 - 2.79iT - 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 0.348iT - 97T^{2} \) |
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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
−10.05915570761694032392784533809, −9.242482731861648161620050514405, −8.808279243708406821645849269772, −8.092360327589428811281223476376, −6.40929982086269333204746041654, −5.37446013733709513358401327963, −4.90787813651172564238040722425, −4.09847792879907851103873961561, −2.89241877728842826427248721066, −2.40646029926149221049358069514,
0.26659252343118219712424611478, 1.29585052507668996924407392497, 2.73115465683998882377577894182, 3.83800649250653800171664323632, 5.38982971557718080020394123414, 5.79273046582908272071881493121, 6.95884258986020185082767463218, 7.57073546366570300009396269922, 7.71420886883109139907806033150, 8.587958861881469159171045101917
