L(s) = 1 | + (−1.22 + 1.22i)2-s − 1.99i·4-s + (1.22 + 1.22i)8-s − i·9-s − 11-s − 0.999·16-s + (1.22 + 1.22i)18-s + (1.22 − 1.22i)22-s + (−1.22 − 1.22i)23-s − i·29-s − 1.99·36-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯ |
L(s) = 1 | + (−1.22 + 1.22i)2-s − 1.99i·4-s + (1.22 + 1.22i)8-s − i·9-s − 11-s − 0.999·16-s + (1.22 + 1.22i)18-s + (1.22 − 1.22i)22-s + (−1.22 − 1.22i)23-s − i·29-s − 1.99·36-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3897577134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3897577134\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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
−9.741500455314076586857271922253, −8.888263754745299180339402948652, −8.172957138962982599987499769013, −7.58369017345640506555482148653, −6.59709413734572691190127769182, −6.06864411573423634846182634193, −5.15318730674635754156384285009, −3.87326402879235251066294548054, −2.28612800923728337838285302019, −0.50897020455499073638655761821,
1.53842539097858420689547349060, 2.49933563134259233916872440427, 3.39516802043105116080282867516, 4.68077485908352453628953452659, 5.70432737776668382567047966735, 7.12911639013463136000110295451, 8.126146910770123734530555379863, 8.175639597153339557932757874344, 9.480840996255004796684298910741, 9.992927765288911876168428299009