L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + 2i·13-s − 1.00·16-s + 1.00·18-s + (−0.707 + 0.707i)25-s + (1.41 − 1.41i)26-s + (0.707 + 0.707i)32-s + (−0.707 − 0.707i)36-s + (0.707 + 0.707i)49-s + 1.00·50-s − 2.00·52-s + (1.41 + 1.41i)53-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + 2i·13-s − 1.00·16-s + 1.00·18-s + (−0.707 + 0.707i)25-s + (1.41 − 1.41i)26-s + (0.707 + 0.707i)32-s + (−0.707 − 0.707i)36-s + (0.707 + 0.707i)49-s + 1.00·50-s − 2.00·52-s + (1.41 + 1.41i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6047649598\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6047649598\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + (0.707 - 0.707i)T^{2} \) |
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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
−10.09035560041178367151190129800, −9.145525200130693397355686146848, −8.783512828484458317585369240634, −7.72128157418238508616334322503, −7.05791250135610464860817171784, −5.96628239236071102952080350543, −4.68533095736488509834066282519, −3.84080170621645936810794188847, −2.59063033828554983524913583898, −1.68419321045449156319243582591,
0.68720107346924350493028983890, 2.47777126484619914499070930457, 3.69877674528110823115230203257, 5.16635277011696563143523971531, 5.77285885984131962506888804467, 6.56293357944531347313049841148, 7.59430638131393476972158937878, 8.265446210646297376629930137086, 8.889634689569386908436472950085, 9.915569738276304808931199629022