L(s) = 1 | + (−1.16 + 2.57i)2-s + (−5.29 − 5.99i)4-s + 0.300i·5-s + 7i·7-s + (21.6 − 6.70i)8-s + (−0.774 − 0.348i)10-s + 9.71·11-s − 76.4·13-s + (−18.0 − 8.13i)14-s + (−7.83 + 63.5i)16-s − 93.2i·17-s − 81.8i·19-s + (1.79 − 1.59i)20-s + (−11.2 + 25.0i)22-s + 185.·23-s + ⋯ |
L(s) = 1 | + (−0.410 + 0.911i)2-s + (−0.662 − 0.749i)4-s + 0.0268i·5-s + 0.377i·7-s + (0.955 − 0.296i)8-s + (−0.0244 − 0.0110i)10-s + 0.266·11-s − 1.63·13-s + (−0.344 − 0.155i)14-s + (−0.122 + 0.992i)16-s − 1.33i·17-s − 0.988i·19-s + (0.0201 − 0.0177i)20-s + (−0.109 + 0.242i)22-s + 1.68·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.060739540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060739540\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 - 2.57i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 0.300iT - 125T^{2} \) |
| 11 | \( 1 - 9.71T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 81.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 139. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 30.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 173. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 356. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 154. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 167.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 403. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 156.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 565.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 542. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 809. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 192.T + 9.12e5T^{2} \) |
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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
−11.51242952427921573955822222372, −10.36129912675028316463859598863, −9.335177945453991598225031432832, −8.796497048522980192156900713707, −7.30251522778971454305450321160, −6.92558490985855410549145816958, −5.35725575187955262657692385196, −4.68809339867224046960784806159, −2.64006324046183594242661076025, −0.55735370719762373170180062699,
1.22569918813528617210505539583, 2.73062751371421984885217073017, 4.03406239192333462767889696032, 5.15469773005828087763275242620, 6.89448444323433167646047756643, 7.88133080471272011356611171702, 8.917476775163298146669537061203, 9.884411256906405349005610652913, 10.61478782764226406598211262846, 11.54461971719040465564585438116