L(s) = 1 | − i·3-s − 9-s + (−1 + i)11-s + 2i·17-s + (−1 + i)19-s + i·25-s + i·27-s + (1 + i)33-s + (−1 − i)43-s + 49-s + 2·51-s + (1 + i)57-s + (1 − i)59-s + (−1 + i)67-s + 75-s + ⋯ |
L(s) = 1 | − i·3-s − 9-s + (−1 + i)11-s + 2i·17-s + (−1 + i)19-s + i·25-s + i·27-s + (1 + i)33-s + (−1 − i)43-s + 49-s + 2·51-s + (1 + i)57-s + (1 − i)59-s + (−1 + i)67-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7191511240\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7191511240\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (1 - i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.615127789385884225835831366804, −8.334956531924677402425154335216, −7.48632183372578108001221370051, −6.90524386471279579507214698368, −5.98098461228239769989367787400, −5.47381964105356767147923368354, −4.32599727491055933842671294216, −3.40131557006202682967208469783, −2.17775377867540877623160208059, −1.62655482113198672143716053480,
0.41482448358526533172098208651, 2.60127571082889161222485459384, 2.94330913149798377784932212865, 4.17118659043604906272646755827, 4.88667237445933256088071658192, 5.49873122030689284311615634810, 6.37950637268068565718746483443, 7.28313698737298633924444932258, 8.254672618098637174700978003110, 8.748587718119645416211023497258