L(s) = 1 | + i·3-s − 9-s + 2i·11-s + 25-s − i·27-s − 2·33-s − 49-s − 2i·59-s + 2·73-s + i·75-s + 81-s − 2i·83-s + 2·97-s − 2i·99-s − 2i·107-s + ⋯ |
L(s) = 1 | + i·3-s − 9-s + 2i·11-s + 25-s − i·27-s − 2·33-s − 49-s − 2i·59-s + 2·73-s + i·75-s + 81-s − 2i·83-s + 2·97-s − 2i·99-s − 2i·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9082539329\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9082539329\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 2iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 2T + T^{2} \) |
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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
−10.57579604182470278989178742714, −9.842610997450187228092650115146, −9.309281712207652705462010343223, −8.296824494016569845828844685206, −7.29631093228283246496110921738, −6.35265223941422036836609645592, −5.04220861762749904743178176906, −4.56634549720434133804464441339, −3.41007626517704594129377904426, −2.11411251405953769947321261844,
1.02777572148403383408704263892, 2.63984763760398080177827851398, 3.56565119294518014715373598869, 5.18517096887695521932682548074, 6.04541178656737113000583711432, 6.74423643649784553539034866123, 7.83229911155073064036663204386, 8.503231277316214331094544444154, 9.183908354581915427311547196559, 10.58411748633036289322809683421