L(s) = 1 | + (−0.285 + 0.495i)2-s + (0.836 + 1.44i)4-s + (0.714 − 1.23i)7-s − 2.10·8-s + (1.33 − 2.31i)11-s + (2.33 + 4.04i)13-s + (0.408 + 0.707i)14-s + (−1.07 + 1.85i)16-s + 2.67·17-s + 4.67·19-s + (0.764 + 1.32i)22-s + (2.95 + 5.12i)23-s − 2.67·26-s + 2.38·28-s + (−4.74 + 8.21i)29-s + ⋯ |
L(s) = 1 | + (−0.202 + 0.350i)2-s + (0.418 + 0.724i)4-s + (0.269 − 0.467i)7-s − 0.742·8-s + (0.402 − 0.697i)11-s + (0.648 + 1.12i)13-s + (0.109 + 0.189i)14-s + (−0.267 + 0.464i)16-s + 0.648·17-s + 1.07·19-s + (0.162 + 0.282i)22-s + (0.616 + 1.06i)23-s − 0.524·26-s + 0.451·28-s + (−0.881 + 1.52i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28538 + 0.908352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28538 + 0.908352i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.285 - 0.495i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.714 + 1.23i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.33 - 4.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + (-2.95 - 5.12i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.74 - 8.21i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.48 + 6.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 + (0.735 + 1.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.235 + 0.408i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.47 - 6.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + (0.571 + 0.990i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.26 + 2.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.29 + 5.70i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 + (-0.143 + 0.249i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.14 + 3.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (-3.91 + 6.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.04037013250849574621302076097, −9.465215213945976515276710488163, −8.988471055499903159457907745028, −7.86922721177539388994256228431, −7.29607535368830955247128006003, −6.39133630326176267375337801986, −5.39227794385416358831091265240, −3.91279426596856913573648175749, −3.22122422569522911619692401394, −1.49364915273591953502380306065,
1.03262825331258798862399361285, 2.32630054808145339630130504395, 3.49258447037491906021984847057, 5.06038852451021414638018842775, 5.73590099541604399322310653752, 6.73009801683293069030038149715, 7.74822216342740052450693086934, 8.758740345126331376872897295566, 9.652110827173194775699760142826, 10.29368997413160756989184078883