L(s) = 1 | − i·2-s + 0.732·3-s − 4-s + i·5-s − 0.732i·6-s − 3i·7-s + i·8-s − 2.46·9-s + 10-s + 3i·11-s − 0.732·12-s − 3·14-s + 0.732i·15-s + 16-s + 8.19·17-s + 2.46i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.422·3-s − 0.5·4-s + 0.447i·5-s − 0.298i·6-s − 1.13i·7-s + 0.353i·8-s − 0.821·9-s + 0.316·10-s + 0.904i·11-s − 0.211·12-s − 0.801·14-s + 0.189i·15-s + 0.250·16-s + 1.98·17-s + 0.580i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.809597314\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.809597314\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 - 8.19T + 17T^{2} \) |
| 19 | \( 1 - 0.464iT - 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 4.73iT - 31T^{2} \) |
| 37 | \( 1 + 0.803iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 6.19T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 - 8.19iT - 83T^{2} \) |
| 89 | \( 1 - 6.80iT - 89T^{2} \) |
| 97 | \( 1 + 9.12iT - 97T^{2} \) |
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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
−9.419864888436732520964596023013, −8.456490261052811752533669474788, −7.52190768901412373426461879125, −7.13761377512085260206675851467, −5.78401328389431871716137649711, −4.94709269719712944372200773376, −3.77291081393014463298100336684, −3.25483772736036692600887075076, −2.16249033421601945887157645653, −0.820473191830407102326357814713,
1.09840629621767469319677331651, 2.83865023905981880684025848173, 3.37820204717216844674189582541, 4.89223789236982260006932333285, 5.58228704271219876475837184598, 6.02609855588364709803153131556, 7.25792246566983425657103817841, 8.096176103633895199041021969351, 8.704011235053005200957805283674, 9.114989242058518012118485968504