| L(s) = 1 | − 23.5i·3-s + 39.5·7-s − 311.·9-s − 236. i·11-s + 942. i·13-s + 1.10e3·17-s − 2.31e3i·19-s − 931. i·21-s − 861.·23-s + 1.61e3i·27-s + 1.66e3i·29-s + 4.18e3·31-s − 5.57e3·33-s − 1.45e4i·37-s + 2.21e4·39-s + ⋯ |
| L(s) = 1 | − 1.51i·3-s + 0.305·7-s − 1.28·9-s − 0.589i·11-s + 1.54i·13-s + 0.925·17-s − 1.47i·19-s − 0.461i·21-s − 0.339·23-s + 0.425i·27-s + 0.368i·29-s + 0.782·31-s − 0.890·33-s − 1.75i·37-s + 2.33·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5687131389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5687131389\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 23.5iT - 243T^{2} \) |
| 7 | \( 1 - 39.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 236. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 942. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.31e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 861.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.66e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.24e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.24e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.11e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.78e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.45e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
| 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
−8.723502771756559658384076712918, −8.023613169491205488488731447250, −7.06053189939733005131728839800, −6.61893917018429924547719368987, −5.60516191491597011188485550668, −4.47794576302888146482866505848, −3.10238493523995602253188313004, −2.00867654097811933792293392396, −1.21614302252411029040952444395, −0.11431322458028902680481745703,
1.47040976892409855037365698912, 3.06531178259580022673721020604, 3.69802176555127879029023995904, 4.84129276072936338556248984361, 5.34991922764009220064335656597, 6.41047140747676049140784065718, 7.974943721608770556263635021743, 8.222615254103444584474437475241, 9.623106470800157907978362933682, 10.10612045206817537193754404989
