L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s − 1.41·11-s − 1.00i·15-s − 1.41i·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s − 1.41·29-s + (1.00 − 1.00i)33-s − 1.41i·35-s + (0.707 + 0.707i)45-s − i·49-s + (1.41 + 1.41i)53-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s − 1.41·11-s − 1.00i·15-s − 1.41i·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s − 1.41·29-s + (1.00 − 1.00i)33-s − 1.41i·35-s + (0.707 + 0.707i)45-s − i·49-s + (1.41 + 1.41i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1418964997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1418964997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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
−8.643551936656958836002035600918, −7.73836138917316169352915302856, −7.01631656091592408685180313223, −6.16147523370628848105410247221, −5.63664737673803762005606925445, −4.86905661455494683552333515616, −3.83314349361459899363925217196, −3.14601656795937268743991551576, −2.38531818175852806652563158951, −0.11083515603850583116527908948,
0.915053585875729798530667157335, 2.28597611445949993884938697453, 3.44411546674206746126332692837, 4.24776924357716495338402229690, 5.18956083554128354429961786976, 5.71109969180679611295955930872, 6.76965564723351896655981742295, 7.32053554461701621740955088382, 7.84894924193560569017238493010, 8.567590256999021496499714759211