L(s) = 1 | + (0.157 + 0.588i)2-s − i·3-s + (1.41 − 0.814i)4-s + (0.529 − 1.97i)5-s + (0.588 − 0.157i)6-s + (−2.23 − 1.40i)7-s + (1.56 + 1.56i)8-s − 9-s + 1.24·10-s + (0.0718 + 0.0718i)11-s + (−0.814 − 1.41i)12-s + (−3.09 − 1.85i)13-s + (0.476 − 1.53i)14-s + (−1.97 − 0.529i)15-s + (0.955 − 1.65i)16-s + (1.81 + 3.14i)17-s + ⋯ |
L(s) = 1 | + (0.111 + 0.416i)2-s − 0.577i·3-s + (0.705 − 0.407i)4-s + (0.236 − 0.884i)5-s + (0.240 − 0.0643i)6-s + (−0.846 − 0.532i)7-s + (0.552 + 0.552i)8-s − 0.333·9-s + 0.394·10-s + (0.0216 + 0.0216i)11-s + (−0.235 − 0.407i)12-s + (−0.858 − 0.513i)13-s + (0.127 − 0.411i)14-s + (−0.510 − 0.136i)15-s + (0.238 − 0.413i)16-s + (0.440 + 0.762i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31325 - 0.717170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31325 - 0.717170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.23 + 1.40i)T \) |
| 13 | \( 1 + (3.09 + 1.85i)T \) |
good | 2 | \( 1 + (-0.157 - 0.588i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.529 + 1.97i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.0718 - 0.0718i)T + 11iT^{2} \) |
| 17 | \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.02 + 1.02i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.16 - 2.40i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.79 - 6.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.47 + 1.73i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.94 + 0.788i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.872 + 3.25i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (7.21 + 4.16i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.97 + 0.529i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (5.27 - 9.14i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.16 + 0.581i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 5.95iT - 61T^{2} \) |
| 67 | \( 1 + (4.43 - 4.43i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.0733 + 0.273i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 4.17i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.01 + 8.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.74 + 3.74i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.10 + 7.85i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (16.5 - 4.43i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.98940125316577028632904582160, −10.71517445231759663723659501394, −9.942131184814228960290285440920, −8.758875821373883174891710964172, −7.60120421830077377617148335708, −6.81536329519697141992474080691, −5.83683656177834001299794659525, −4.82895219303694155046164585522, −2.92607064790278203389798407828, −1.22416955358290875311378235961,
2.55430801972867127521205386125, 3.15615498233263845602134778162, 4.67683642743129945206826204787, 6.30284380282536115183042879173, 6.88826494290116984654426669745, 8.200013436821839668924327089393, 9.662834329290720831645448424423, 10.09408188068790004491253451999, 11.18392436096943346914416527264, 11.90562801376738985978851358108