L(s) = 1 | + (0.654 − 0.755i)2-s + (1.12 + 1.50i)3-s + (−0.142 − 0.989i)4-s + (1.86 + 0.133i)6-s + (−0.841 − 0.540i)8-s + (−0.707 + 2.40i)9-s + (−0.909 + 0.584i)11-s + (1.32 − 1.32i)12-s + (−0.959 + 0.281i)16-s + (0.989 − 0.857i)17-s + (1.35 + 2.11i)18-s + (−0.936 − 1.71i)19-s + (−0.153 + 1.07i)22-s + (−0.133 − 1.86i)24-s + (−0.415 − 0.909i)25-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (1.12 + 1.50i)3-s + (−0.142 − 0.989i)4-s + (1.86 + 0.133i)6-s + (−0.841 − 0.540i)8-s + (−0.707 + 2.40i)9-s + (−0.909 + 0.584i)11-s + (1.32 − 1.32i)12-s + (−0.959 + 0.281i)16-s + (0.989 − 0.857i)17-s + (1.35 + 2.11i)18-s + (−0.936 − 1.71i)19-s + (−0.153 + 1.07i)22-s + (−0.133 − 1.86i)24-s + (−0.415 − 0.909i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.660114731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.660114731\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
good | 3 | \( 1 + (-1.12 - 1.50i)T + (-0.281 + 0.959i)T^{2} \) |
| 5 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 11 | \( 1 + (0.909 - 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 17 | \( 1 + (-0.989 + 0.857i)T + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.936 + 1.71i)T + (-0.540 + 0.841i)T^{2} \) |
| 23 | \( 1 + (-0.540 + 0.841i)T^{2} \) |
| 29 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 31 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.13 - 0.847i)T + (0.281 + 0.959i)T^{2} \) |
| 43 | \( 1 + (0.0303 - 0.139i)T + (-0.909 - 0.415i)T^{2} \) |
| 47 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 53 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (1.19 - 1.59i)T + (-0.281 - 0.959i)T^{2} \) |
| 61 | \( 1 + (0.755 - 0.654i)T^{2} \) |
| 67 | \( 1 + (0.0801 - 0.557i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (1.74 - 0.512i)T + (0.841 - 0.540i)T^{2} \) |
| 79 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 0.0855i)T + (0.989 + 0.142i)T^{2} \) |
| 97 | \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \) |
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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
−10.53812239294290037946674538063, −9.880143903253433640709461592864, −9.270214420550771274287658797681, −8.416774182045797212114390479630, −7.29368158595115337912888114730, −5.67280509704095964558304346322, −4.71872531035251595220640034897, −4.25794558944904644099003056460, −2.93741114013149529147065641854, −2.46265067207093483053294352210,
1.87780954807291510019875714084, 3.09419888256865580070144159930, 3.87048865811984084505948243017, 5.69189023050477380344924344964, 6.17009360647573282158565578171, 7.33244846555670625812906500404, 7.937076349814035899333734733665, 8.351332041406711903547806309320, 9.326917101682718658191012927613, 10.69627985179855010255984940448