L(s) = 1 | + (−1.41 + 1.41i)5-s + 7-s + (3.97 + 3.97i)11-s + (−1.10 + 1.10i)13-s − 6.39i·17-s + 2.97i·23-s + 0.999i·25-s + (5.53 + 5.53i)29-s − 2.20i·31-s + (−1.41 + 1.41i)35-s + (−1 − i)37-s − 2.08·41-s + (4.68 − 4.68i)43-s + 8.77·47-s + 49-s + ⋯ |
L(s) = 1 | + (−0.632 + 0.632i)5-s + 0.377·7-s + (1.19 + 1.19i)11-s + (−0.305 + 0.305i)13-s − 1.55i·17-s + 0.620i·23-s + 0.199i·25-s + (1.02 + 1.02i)29-s − 0.396i·31-s + (−0.239 + 0.239i)35-s + (−0.164 − 0.164i)37-s − 0.325·41-s + (0.713 − 0.713i)43-s + 1.28·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.668175036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668175036\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 11 | \( 1 + (-3.97 - 3.97i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.10 - 1.10i)T - 13iT^{2} \) |
| 17 | \( 1 + 6.39iT - 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 - 2.97iT - 23T^{2} \) |
| 29 | \( 1 + (-5.53 - 5.53i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.20iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + (-4.68 + 4.68i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 + (3.83 - 3.83i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.51 - 9.51i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.10 - 5.10i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.10 + 6.10i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.62iT - 71T^{2} \) |
| 73 | \( 1 + 7.04iT - 73T^{2} \) |
| 79 | \( 1 - 4.41iT - 79T^{2} \) |
| 83 | \( 1 + (-10.0 + 10.0i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.774234775565750235177432263826, −7.54656939940252281383104616861, −7.23352287170948288195089402283, −6.74385394356697677262149839666, −5.61849561429820152118637805997, −4.70887694330357294410558934432, −4.14099986344046880489154402036, −3.20360892012347565769050749661, −2.27395290560899950717877051125, −1.14891882500326072162677785212,
0.55696138102526987282621870570, 1.48948620040732756206934720785, 2.78503671848845591791711496225, 3.88485224750827175009847623293, 4.25413319485893721326317203962, 5.24520751379263464423071931609, 6.14940151782878690319331497776, 6.61820232261092876753512328091, 7.79833631765472822044877221716, 8.396196577073959741743755854760