L(s) = 1 | + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s − 1.41i·12-s − 1.41i·13-s − 0.999·16-s − 1.41i·18-s + 19-s + 2.00·26-s − 1.41i·32-s + 1.00·36-s − 1.41i·37-s + 1.41i·38-s + 2.00·39-s + ⋯ |
L(s) = 1 | + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s − 1.00·9-s − 1.41i·12-s − 1.41i·13-s − 0.999·16-s − 1.41i·18-s + 19-s + 2.00·26-s − 1.41i·32-s + 1.00·36-s − 1.41i·37-s + 1.41i·38-s + 2.00·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8608239639\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8608239639\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−11.37990582308993058538540547590, −10.57623719591102067851626240796, −9.718099286941911623165117393099, −8.920466863220392946068575919118, −7.994480307557649378855852847443, −7.17561271918665208873725262681, −5.81023703002536076086294186153, −5.30219161385696357359246288479, −4.29637028922848141649342123691, −3.05790241124699620181144792259,
1.31743840664703523636206152464, 2.22458540386151061200000898513, 3.44325559732078420873714628887, 4.77822955115321017582691481289, 6.38299027142892247989561511415, 7.03731933014322027177403202566, 8.111811888096179358965101859191, 9.217191113077171857790777816209, 9.975729415498283986647053493887, 11.17766425103694944005914721705