L(s) = 1 | + (1 − i)2-s − 4i·3-s − 2i·4-s + (3 + 3i)5-s + (−4 − 4i)6-s − 4·7-s + (−2 − 2i)8-s − 7·9-s + 6·10-s − 4i·11-s − 8·12-s + (3 + 3i)13-s + (−4 + 4i)14-s + (12 − 12i)15-s − 4·16-s + (23 + 23i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s − 1.33i·3-s − 0.5i·4-s + (0.600 + 0.600i)5-s + (−0.666 − 0.666i)6-s − 0.571·7-s + (−0.250 − 0.250i)8-s − 0.777·9-s + 0.600·10-s − 0.363i·11-s − 0.666·12-s + (0.230 + 0.230i)13-s + (−0.285 + 0.285i)14-s + (0.800 − 0.800i)15-s − 0.250·16-s + (1.35 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.10398 - 1.19913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10398 - 1.19913i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 37 | \( 1 - 37iT \) |
good | 3 | \( 1 + 4iT - 9T^{2} \) |
| 5 | \( 1 + (-3 - 3i)T + 25iT^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 + 4iT - 121T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 169iT^{2} \) |
| 17 | \( 1 + (-23 - 23i)T + 289iT^{2} \) |
| 19 | \( 1 + (-10 - 10i)T + 361iT^{2} \) |
| 23 | \( 1 + (10 + 10i)T + 529iT^{2} \) |
| 29 | \( 1 + (19 - 19i)T - 841iT^{2} \) |
| 31 | \( 1 + (18 - 18i)T - 961iT^{2} \) |
| 41 | \( 1 + 74iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-42 - 42i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 44T + 2.20e3T^{2} \) |
| 53 | \( 1 + 80T + 2.80e3T^{2} \) |
| 59 | \( 1 + (54 + 54i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (3 - 3i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 - 12iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 124T + 5.04e3T^{2} \) |
| 73 | \( 1 - 10iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-14 - 14i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 + 64T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-17 + 17i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-129 - 129i)T + 9.40e3iT^{2} \) |
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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
−13.94971302798630805091721976874, −12.80080056819426928365159945434, −12.28436777667639013895495109900, −10.84711011935171845950599526817, −9.776008111707046971156775444856, −8.032794725847557928015040339350, −6.61779079946939470987437219179, −5.83275302681272204421359771488, −3.33921956009323824632251740169, −1.65956295605923317123107545788,
3.37875699457624295053014397407, 4.84866898617643768664888855560, 5.76186677957676864048926210040, 7.56444323665446958086294095506, 9.401469888143138558411485371251, 9.680052062216690319669824061045, 11.29772222717272103930219538802, 12.65572488415663634665699921647, 13.67807641372599725004753761468, 14.74582601661900919183475078088