L(s) = 1 | + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s + 16i·7-s − 8i·8-s − 9·9-s + 12·11-s + 12i·12-s + 38i·13-s − 32·14-s + 16·16-s + 126i·17-s − 18i·18-s − 20·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.863i·7-s − 0.353i·8-s − 0.333·9-s + 0.328·11-s + 0.288i·12-s + 0.810i·13-s − 0.610·14-s + 0.250·16-s + 1.79i·17-s − 0.235i·18-s − 0.241·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.654794 + 1.05948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.654794 + 1.05948i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 16iT - 343T^{2} \) |
| 11 | \( 1 - 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 126iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 168iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 30T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 42T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 96iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 198iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 660T + 2.05e5T^{2} \) |
| 61 | \( 1 + 538T + 2.26e5T^{2} \) |
| 67 | \( 1 + 884iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 792T + 3.57e5T^{2} \) |
| 73 | \( 1 - 218iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 520T + 4.93e5T^{2} \) |
| 83 | \( 1 + 492iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−12.88935979540794446194687714540, −12.11151908684600446439981304031, −10.97001335860802327418590291120, −9.431225990465431248687162928918, −8.625501998755661139369293637728, −7.55286991847974058170407434075, −6.37498172551979777044192897346, −5.54244122201709576918154188481, −3.85685864719197492741733155384, −1.82200097105262489841901868116,
0.61762565942453821655206538542, 2.80293928994829497994758352887, 4.11737922925898063725803719450, 5.20018035181641026023594236260, 6.89055927304721360609988287331, 8.242080266763142198116972280680, 9.421910594330110596214688032552, 10.28158431250180428078133453845, 11.07629486055602669308441468570, 12.07742495508161353372830052660