| L(s) = 1 | + 14·7-s − 28·13-s + 16·19-s + 70·31-s − 88·37-s + 44·43-s + 49·49-s + 40·61-s − 28·67-s − 178·73-s + 220·79-s − 392·91-s − 22·97-s + 44·103-s − 104·109-s + 161·121-s + 127-s + 131-s + 224·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 250·169-s + ⋯ |
| L(s) = 1 | + 2·7-s − 2.15·13-s + 0.842·19-s + 2.25·31-s − 2.37·37-s + 1.02·43-s + 49-s + 0.655·61-s − 0.417·67-s − 2.43·73-s + 2.78·79-s − 4.30·91-s − 0.226·97-s + 0.427·103-s − 0.954·109-s + 1.33·121-s + 0.00787·127-s + 0.00763·131-s + 1.68·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.47·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.950365834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.950365834\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 161 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1358 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1502 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5537 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6638 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5794 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 89 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13049 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15518 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.902023606322581233520802931161, −8.455429641790874219273150028548, −7.84780505131936220589473138449, −7.83038658055845829183459005011, −7.52291835916958128505661889136, −6.97567642222523469654206883170, −6.73713691741009372022070830311, −6.17288129545365564927716158273, −5.55072200278904671603400023127, −5.23681602491675607287701995505, −4.92118015792522538287520009583, −4.65573116259555477882253452401, −4.32592636248504591817219522069, −3.67019446756539167420453474536, −3.05101150205444772106417767667, −2.61025161729083227228363151778, −2.14772681313460150487287598966, −1.65769194951764030790772072954, −1.13277732945560303007754138851, −0.41612008327888039510440002697,
0.41612008327888039510440002697, 1.13277732945560303007754138851, 1.65769194951764030790772072954, 2.14772681313460150487287598966, 2.61025161729083227228363151778, 3.05101150205444772106417767667, 3.67019446756539167420453474536, 4.32592636248504591817219522069, 4.65573116259555477882253452401, 4.92118015792522538287520009583, 5.23681602491675607287701995505, 5.55072200278904671603400023127, 6.17288129545365564927716158273, 6.73713691741009372022070830311, 6.97567642222523469654206883170, 7.52291835916958128505661889136, 7.83038658055845829183459005011, 7.84780505131936220589473138449, 8.455429641790874219273150028548, 8.902023606322581233520802931161
