L(s) = 1 | + 1.73i·7-s − 13-s + 1.73i·19-s − 25-s + 37-s − 1.99·49-s + 61-s + 1.73i·67-s + 73-s − 1.73i·79-s − 1.73i·91-s + 97-s + 1.73i·103-s + 2·109-s + ⋯ |
L(s) = 1 | + 1.73i·7-s − 13-s + 1.73i·19-s − 25-s + 37-s − 1.99·49-s + 61-s + 1.73i·67-s + 73-s − 1.73i·79-s − 1.73i·91-s + 97-s + 1.73i·103-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9477979833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9477979833\) |
\(L(1)\) |
|
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 | \( 1 - 1.73iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 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
−9.701759286508287861707756639335, −8.933819712347643075619844316162, −8.159215719417198043085060233607, −7.50737926119342817179520179736, −6.26784205467010861434491546022, −5.72506125508415380321200066592, −4.96311573602473579778484235358, −3.80139067778985993802534900300, −2.65943885781688702367402266851, −1.88351757240761996605787485158,
0.71002701286702548198936457465, 2.26899311024382299705108684468, 3.45412206338884521975917905470, 4.39354425764661822369255022095, 4.98734926646215308428529286052, 6.26522248990561709667051410290, 7.14788637360418958420857473214, 7.49363298184073124659858536979, 8.443455406363999126001194511581, 9.615462340917795172134851838264