L(s) = 1 | − i·2-s + i·3-s − 4-s + 0.700i·5-s + 6-s + i·8-s − 9-s + 0.700·10-s + (0.176 − 3.31i)11-s − i·12-s + 7.03·13-s − 0.700·15-s + 16-s − 0.617·17-s + i·18-s + 0.783·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.313i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.221·10-s + (0.0531 − 0.998i)11-s − 0.288i·12-s + 1.95·13-s − 0.180·15-s + 0.250·16-s − 0.149·17-s + 0.235i·18-s + 0.179·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.779073098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779073098\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.176 + 3.31i)T \) |
good | 5 | \( 1 - 0.700iT - 5T^{2} \) |
| 13 | \( 1 - 7.03T + 13T^{2} \) |
| 17 | \( 1 + 0.617T + 17T^{2} \) |
| 19 | \( 1 - 0.783T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 2.09iT - 29T^{2} \) |
| 31 | \( 1 + 6.17iT - 31T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 + 3.62iT - 43T^{2} \) |
| 47 | \( 1 - 2.34iT - 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 + 3.76iT - 59T^{2} \) |
| 61 | \( 1 - 8.36T + 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 5.28T + 73T^{2} \) |
| 79 | \( 1 - 4.33iT - 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−8.532648665511387187799311259157, −8.290731466798174644317365463079, −6.98338111762034046390017069005, −6.04008886088943761924750412508, −5.57370000651727650905891528666, −4.41939413997056345282318803410, −3.62227094535407060250383067669, −3.18781653122818637208027381549, −1.93324959030994124005688238255, −0.69566157134856679793799980834,
0.997873949149393979476982268714, 1.95505442284027865956862735112, 3.36392219759847027458382498948, 4.17804527268623275404400905797, 5.10268695699761000459490054760, 5.87331701226785854637344656283, 6.63237385734768767846228898774, 7.11175338497711037635316622491, 8.135724590849270128230460217691, 8.518199780092247754833169562558