L(s) = 1 | + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 17-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + 0.999·28-s + (1 − 1.73i)29-s − 0.999·35-s + 0.999·44-s + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 17-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + 0.999·28-s + (1 − 1.73i)29-s − 0.999·35-s + 0.999·44-s + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9000004152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9000004152\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.984144676664577140897230217441, −9.835433203939503953090709192534, −8.949099719176624520828267861193, −7.87142172923039882158240569249, −6.68105614222819842150755975577, −6.09350537486027703388368163867, −5.31468164366994026288209638955, −4.21255227783365436383376621636, −2.83762131276282827919223607362, −1.80642000281342737241429253877,
0.952758889404859546188942638999, 3.00602094776612171978053568942, 3.70209033157564122148272862830, 4.88927751558656183274434647864, 5.64552044558599080666745094203, 6.79834002760177459808336535315, 7.892317866102501250990284685082, 8.384684693344255517241302543645, 9.205925282059142179978326129197, 10.11901048241373687820457123035