L(s) = 1 | + (0.292 + 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + i·13-s + 17-s + (−1 + i)19-s + (−0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (−1.70 − 0.707i)31-s − 2i·47-s + (0.292 − 0.292i)49-s + (−1 − i)59-s + (0.292 + 0.707i)61-s + (−0.292 + 0.707i)63-s + 1.41·67-s + ⋯ |
L(s) = 1 | + (0.292 + 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + i·13-s + 17-s + (−1 + i)19-s + (−0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (−1.70 − 0.707i)31-s − 2i·47-s + (0.292 − 0.292i)49-s + (−1 − i)59-s + (0.292 + 0.707i)61-s + (−0.292 + 0.707i)63-s + 1.41·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.499277176\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499277176\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (1 + i)T + iT^{2} \) |
| 61 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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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
−8.790735454313308133507810841983, −8.266054626820765042242427050347, −7.29820354540385725768237394025, −6.64628932619176871642663951621, −5.82554737287717930316175855053, −5.16986754801424293037742594531, −3.95331049723018105715266706884, −3.71573413621836351189677079384, −2.05049512282581678332565981703, −1.56631331079462135252561265614,
0.987063898612726495354128234983, 1.89165184224154220249585687522, 3.36890111900793522041258081456, 4.01296245533721836591930632368, 4.61468409084732001912786750308, 5.78441157216822722204420176419, 6.45021749863520625541663044392, 7.35409264639173351513302463897, 7.59814388904729068551794702637, 8.775801218066058847178694764028