| L(s) = 1 | + 3i·7-s + 5i·13-s + 5·19-s − 4i·23-s + 4·29-s + 5·31-s − 10i·37-s + 10·41-s + i·43-s − 2i·47-s − 2·49-s − 10i·53-s + 10·59-s − 5·61-s + 3i·67-s + ⋯ |
| L(s) = 1 | + 1.13i·7-s + 1.38i·13-s + 1.14·19-s − 0.834i·23-s + 0.742·29-s + 0.898·31-s − 1.64i·37-s + 1.56·41-s + 0.152i·43-s − 0.291i·47-s − 0.285·49-s − 1.37i·53-s + 1.30·59-s − 0.640·61-s + 0.366i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131847903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131847903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.148468579894745603695875477969, −7.28023168690919572646069183866, −6.63541625678875215429004032080, −5.94568846680347586150238349800, −5.27508510328405042018960804052, −4.50723559878007910352613320723, −3.75262958844375785241169807452, −2.64691516672709656661156680712, −2.17228386581809727127179070735, −0.942848077601048138011088464609,
0.67185663936785838554514551888, 1.34120108166275898542197366711, 2.83177302732851896102600997987, 3.28585768730193686191191002202, 4.25532822294967279801780018616, 4.90875745841107105449303967299, 5.72872715915044863802487067710, 6.37927225341814861272845710688, 7.36774563390800496799461939241, 7.63699792013487946406843668830
