L(s) = 1 | + 1.41·2-s + (−0.707 − 1.58i)3-s + 2.00·4-s + (−1.00 − 2.23i)6-s − 3.16i·7-s + 2.82·8-s + (−2.00 + 2.23i)9-s + (−1.41 − 3.16i)12-s − 4.47i·14-s + 4.00·16-s + (−2.82 + 3.16i)18-s + (−5.00 + 2.23i)21-s + 1.41·23-s + (−2.00 − 4.47i)24-s + (4.94 + 1.58i)27-s − 6.32i·28-s + ⋯ |
L(s) = 1 | + 1.00·2-s + (−0.408 − 0.912i)3-s + 1.00·4-s + (−0.408 − 0.912i)6-s − 1.19i·7-s + 1.00·8-s + (−0.666 + 0.745i)9-s + (−0.408 − 0.912i)12-s − 1.19i·14-s + 1.00·16-s + (−0.666 + 0.745i)18-s + (−1.09 + 0.487i)21-s + 0.294·23-s + (−0.408 − 0.912i)24-s + (0.952 + 0.304i)27-s − 1.19i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73141 - 1.12235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73141 - 1.12235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (0.707 + 1.58i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.16iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.64132590055451221544924785125, −10.99154064805850309928776110131, −10.11501064144395588854236741710, −8.323886409467531876953720789446, −7.25271579290364933410954135621, −6.74966095109870690982018776324, −5.56387658201116029539267847275, −4.48114490032672114408415440880, −3.09350347636820903833734666410, −1.40846540252742740862468046415,
2.46962120675756535733012006642, 3.72870891484257363571015925605, 4.90793919715533224343542719492, 5.70347239274477912542262981235, 6.55760537413229617345144475204, 8.099889564467530799192740993572, 9.253797081645866551219571016603, 10.22210358065822550779569645023, 11.26620435365702918562014884633, 11.86510524643463939715639775320