L(s) = 1 | − i·2-s + 3·3-s + 7·4-s + 7i·5-s − 3i·6-s + 10i·7-s − 15i·8-s + 9·9-s + 7·10-s + 22i·11-s + 21·12-s + 10·14-s + 21i·15-s + 41·16-s − 37·17-s − 9i·18-s + ⋯ |
L(s) = 1 | − 0.353i·2-s + 0.577·3-s + 0.875·4-s + 0.626i·5-s − 0.204i·6-s + 0.539i·7-s − 0.662i·8-s + 0.333·9-s + 0.221·10-s + 0.603i·11-s + 0.505·12-s + 0.190·14-s + 0.361i·15-s + 0.640·16-s − 0.527·17-s − 0.117i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.073958774\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.073958774\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + iT - 8T^{2} \) |
| 5 | \( 1 - 7iT - 125T^{2} \) |
| 7 | \( 1 - 10iT - 343T^{2} \) |
| 11 | \( 1 - 22iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 37T + 4.91e3T^{2} \) |
| 19 | \( 1 - 30iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 162T + 1.21e4T^{2} \) |
| 29 | \( 1 + 113T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 13iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 285iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 246T + 7.95e4T^{2} \) |
| 47 | \( 1 - 462iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 537T + 1.48e5T^{2} \) |
| 59 | \( 1 + 576iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 635T + 2.26e5T^{2} \) |
| 67 | \( 1 - 202iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 805iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 884T + 4.93e5T^{2} \) |
| 83 | \( 1 - 518iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 194iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3iT - 9.12e5T^{2} \) |
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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
−10.79460748587518372048437793175, −9.728178961169051786925102325261, −8.949728984531394534499386743901, −7.73828231743449996037902114865, −6.98619977537798814530031605912, −6.19121548472192415221577139489, −4.77352743291811944145030170546, −3.31624187138806325052107449747, −2.61569970950581122404835494376, −1.50429643173402262795720732483,
0.905421738761332474586948828172, 2.29676760800215134062938286735, 3.46349867047418674221993607443, 4.74329879418714130111604751333, 5.84731868777668843860899627645, 6.95098959712692682577528235478, 7.59288940331421725512631197431, 8.626010499916792556473548319619, 9.259996534925932011471982368967, 10.60112324143922350474529930907