L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·8-s + (−0.707 − 1.22i)10-s − 1.41·13-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + 1.41·20-s + (−0.499 − 0.866i)25-s + (0.707 − 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)40-s − 1.41·41-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·8-s + (−0.707 − 1.22i)10-s − 1.41·13-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + 1.41·20-s + (−0.499 − 0.866i)25-s + (0.707 − 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3533893065\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3533893065\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−9.965702299377031751972798314830, −9.087465231311976125086648002661, −8.089576038383647906862720564081, −7.48347638676569615473871292913, −7.02815258369489530782367059805, −6.11190708894644019578999840838, −5.29401768510124845186256481067, −4.19436994871180207183784261692, −3.26916145146759369928842089433, −1.91104522520550825427703019289,
0.31003323001282393933179788417, 1.71010602305280053563800764311, 2.93786377108900925863387528571, 3.93631867236240465507806718378, 4.83802481538778707204668618847, 5.34344098036039832425231811150, 7.11447671973884288167442989895, 7.65654831200167262863158970697, 8.351754668431760920939286560386, 9.276259184125496606630298156791