L(s) = 1 | + i·2-s − 3-s − 4-s + 0.339i·5-s − i·6-s − i·8-s + 9-s − 0.339·10-s − 0.660i·11-s + 12-s + (−0.660 + 3.54i)13-s − 0.339i·15-s + 16-s + 6.54·17-s + i·18-s − 6.08i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.151i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s − 0.107·10-s − 0.199i·11-s + 0.288·12-s + (−0.183 + 0.983i)13-s − 0.0877i·15-s + 0.250·16-s + 1.58·17-s + 0.235i·18-s − 1.39i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.108511843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108511843\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.660 - 3.54i)T \) |
good | 5 | \( 1 - 0.339iT - 5T^{2} \) |
| 11 | \( 1 + 0.660iT - 11T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 6.08iT - 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 3.54iT - 37T^{2} \) |
| 41 | \( 1 + 0.864iT - 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 3.90iT - 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 7.44iT - 67T^{2} \) |
| 71 | \( 1 + 0.884iT - 71T^{2} \) |
| 73 | \( 1 + 10.0iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 16.2iT - 83T^{2} \) |
| 89 | \( 1 - 4.45iT - 89T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−8.422925948141501865414416973958, −7.50054767248647420629476155169, −7.03170836624654333692963086834, −6.23306453176640235193862677709, −5.60724730918288292292424826175, −4.82899899728242885201241157827, −4.08460192359508801677620501846, −3.10283259333111955141840565246, −1.79088174590001586094171823003, −0.44261513362655720622708959670,
0.938517883156279260429224591638, 1.85136676800952173722273364426, 3.10687700509933242271827901417, 3.77088947620707027483228491265, 4.72392538057675040990516380706, 5.59868129708196646077605080860, 5.92221182239081727473727861719, 7.15041169645146759406839030991, 7.943368576912718493695187384763, 8.379819120063533967981409368641