L(s) = 1 | + i·2-s + 4-s + (−1 − 2i)5-s − i·7-s + 3i·8-s + (2 − i)10-s + 6·11-s + 2i·13-s + 14-s − 16-s − 4i·17-s + 6·19-s + (−1 − 2i)20-s + 6i·22-s + (−3 + 4i)25-s − 2·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.5·4-s + (−0.447 − 0.894i)5-s − 0.377i·7-s + 1.06i·8-s + (0.632 − 0.316i)10-s + 1.80·11-s + 0.554i·13-s + 0.267·14-s − 0.250·16-s − 0.970i·17-s + 1.37·19-s + (−0.223 − 0.447i)20-s + 1.27i·22-s + (−0.600 + 0.800i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49653 + 0.353283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49653 + 0.353283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 + iT \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−11.68149172021698514537307669320, −11.15682754621204080457739790913, −9.448295105735178540927949522834, −8.960783065017993763415907324230, −7.61379270299440107768625972683, −7.07117570942000618842025239364, −5.91216171115806701944249425884, −4.78965586740336360807090747123, −3.57134491131064569823169274652, −1.48127063129262097043069794043,
1.63786604545577669190033940755, 3.18346466442169681575130094017, 3.88583663336720952404155266403, 5.84277041949752518384985221923, 6.77502665514268466800942486071, 7.59570723637057592816899295658, 8.988335577470060659572922989961, 9.958907356148204600212707497163, 10.83782949321453739281959390784, 11.65900402784438336251184378974