L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (−0.366 + 0.366i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + 1.93i·11-s − 0.517·14-s − 1.00·16-s + (0.258 − 0.965i)20-s + (−1.36 + 1.36i)22-s + (0.866 + 0.499i)25-s + (−0.366 − 0.366i)28-s − 1.41·29-s + 1.73·31-s + (−0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (−0.366 + 0.366i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + 1.93i·11-s − 0.517·14-s − 1.00·16-s + (0.258 − 0.965i)20-s + (−1.36 + 1.36i)22-s + (0.866 + 0.499i)25-s + (−0.366 − 0.366i)28-s − 1.41·29-s + 1.73·31-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.074560683\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074560683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
good | 7 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 11 | \( 1 - 1.93iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - 1.73T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
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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
−10.31364314604517242656487371977, −9.372777390441916896112974701392, −8.560869696768312381928317825740, −7.61255351026855775669703073984, −7.15639693244075457165838028171, −6.21945964292962409430231108128, −5.05473069616885833400471395787, −4.43315291479376740280265671827, −3.53763696552914733292390171304, −2.28600652316790024360033359782,
0.798149555223504654757891523591, 2.72438323858210132610472295477, 3.52851301491572106562350708004, 4.18201328221036953870092019002, 5.42323024189406418632760598397, 6.23601799066080206632784356162, 7.12014964825880209359349050988, 8.237514367683037813822279016038, 8.966111381566469968714892893339, 10.10771285737255884372209477110