L(s) = 1 | + (−1 − i)11-s − 13-s − 2i·23-s + i·25-s + (−1 − i)37-s + (−1 − i)47-s − i·49-s + (−1 − i)59-s + (−1 + i)71-s + (1 + i)73-s + (1 − i)83-s + (1 − i)97-s + 2·107-s + (−1 + i)109-s + ⋯ |
L(s) = 1 | + (−1 − i)11-s − 13-s − 2i·23-s + i·25-s + (−1 − i)37-s + (−1 − i)47-s − i·49-s + (−1 − i)59-s + (−1 + i)71-s + (1 + i)73-s + (1 − i)83-s + (1 − i)97-s + 2·107-s + (−1 + i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6813400115\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6813400115\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + 2iT - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (1 + i)T + iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (1 - i)T - iT^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−8.479114707396776627801895371896, −7.74654126851050169345744502390, −7.01610783570122534131349390971, −6.24097908528318041254709554502, −5.31066003577159365900450029141, −4.85934667144354491399694721526, −3.71101086828011157979238802297, −2.86597792640677678563461773135, −2.03471703486190963481334392981, −0.36287447309263931697076082015,
1.61027865595398589909353528693, 2.54752532615215223820573898486, 3.40484264466651294328229669767, 4.61373105437148096088441184588, 5.00830964200502202475707942690, 5.92636738845667807504939595264, 6.82294545154454196974248832912, 7.67715184209909167718325525339, 7.85884678801776605512345357631, 9.060915216984344029401056407430