L(s) = 1 | + i·7-s + 1.73i·13-s − 1.73·19-s + 25-s + 2i·31-s − 1.73i·37-s + 1.73i·61-s + 1.73·67-s + 73-s − i·79-s − 1.73·91-s − 97-s − i·103-s + ⋯ |
L(s) = 1 | + i·7-s + 1.73i·13-s − 1.73·19-s + 25-s + 2i·31-s − 1.73i·37-s + 1.73i·61-s + 1.73·67-s + 73-s − i·79-s − 1.73·91-s − 97-s − i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.011134862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011134862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.430317359311620479077130064331, −8.838591704803721334924446271939, −8.429152407224379722889187289674, −7.06299988664014604699404275017, −6.59155415689068975832667498616, −5.65695255260783265523115516377, −4.72147959048422890313670555702, −3.91197042016979219128012983439, −2.60489454967546193983758697352, −1.77589898318638160448823422527,
0.78014142159824132685990337536, 2.36954644930763375406923510681, 3.45381103216667111422212066484, 4.33032740869649264700704894474, 5.21838026674552820286354839953, 6.23812807332223789270393263036, 6.91502407249148196331712360437, 8.018635026747336567420125322454, 8.230105973918683744779069682313, 9.473058775742837307101360930054