L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 7-s − 0.999·8-s − 9-s − 1.73i·11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−1.49 − 0.866i)22-s − 23-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (0.499 + 0.866i)32-s + (0.499 + 0.866i)36-s + 1.73i·37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 7-s − 0.999·8-s − 9-s − 1.73i·11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−1.49 − 0.866i)22-s − 23-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (0.499 + 0.866i)32-s + (0.499 + 0.866i)36-s + 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7493030579\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7493030579\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.493651268341040971641532956191, −8.704452215552103367009069110378, −8.044919118323763129273374989878, −6.37055667514839173299920024060, −6.06148820894904689528403539727, −5.20451022881704267442759786319, −3.85084614058781189252424500176, −3.24363039444552861579197778460, −2.36349393864433788223460608726, −0.49418645872110009508556534167,
2.37007642889111239523378475339, 3.40098446496291043402696903894, 4.33499180401528034990351055655, 5.29672123352404281747730163147, 6.07680643003824312163113423068, 6.89467839796931541184747272690, 7.50665489231399357733332891726, 8.469937811031851222126383239222, 9.303160052920402604799326578851, 9.843954593408458694068303863593