L(s) = 1 | + (0.766 + 0.642i)5-s + (0.342 + 0.939i)9-s + (−0.342 + 0.939i)13-s + (−1.20 + 0.439i)17-s + (0.173 + 0.984i)25-s + (0.816 − 0.218i)29-s + (−0.939 − 0.342i)37-s + (−0.439 + 1.20i)41-s + (−0.342 + 0.939i)45-s + (−0.984 − 0.173i)49-s + (−0.0451 − 0.515i)53-s + (1.64 − 0.766i)61-s + (−0.866 + 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.766 + 0.642i)81-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)5-s + (0.342 + 0.939i)9-s + (−0.342 + 0.939i)13-s + (−1.20 + 0.439i)17-s + (0.173 + 0.984i)25-s + (0.816 − 0.218i)29-s + (−0.939 − 0.342i)37-s + (−0.439 + 1.20i)41-s + (−0.342 + 0.939i)45-s + (−0.984 − 0.173i)49-s + (−0.0451 − 0.515i)53-s + (1.64 − 0.766i)61-s + (−0.866 + 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.766 + 0.642i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.311484965\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311484965\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.939 + 0.342i)T \) |
good | 3 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 7 | \( 1 + (0.984 + 0.173i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.816 + 0.218i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.0451 + 0.515i)T + (-0.984 + 0.173i)T^{2} \) |
| 59 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 61 | \( 1 + (-1.64 + 0.766i)T + (0.642 - 0.766i)T^{2} \) |
| 67 | \( 1 + (-0.984 - 0.173i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 + (0.984 + 0.173i)T^{2} \) |
| 83 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 89 | \( 1 + (0.173 + 1.98i)T + (-0.984 + 0.173i)T^{2} \) |
| 97 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.124327376005589681981559444880, −8.385335083096028687311387146519, −7.50178154754754792409883261711, −6.68629391259267889602627522822, −6.31218291164998781695117588875, −5.12527658526776176655511040584, −4.58351529950196284383368648538, −3.46518937587832876619450212234, −2.29932715202899270116373944516, −1.80077700636066922378148926124,
0.806386495442069763455938371966, 2.04561414614780268845462574423, 3.04264022763408933303666116816, 4.11479688792705774909607988400, 4.97208602100862139410499988551, 5.62591886743066498114785810712, 6.56837461940898213712567600285, 7.04944628721077734589331546444, 8.232552547904586294402023349777, 8.783661860868405901258066792857