L(s) = 1 | + (0.965 + 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + 6-s + (−1.22 − 0.707i)7-s + (0.707 + 0.707i)8-s + 10-s + (1 + i)11-s + (0.965 + 0.258i)12-s + (0.258 + 0.965i)13-s + (−0.999 − i)14-s + (0.866 − 0.499i)15-s + (0.500 + 0.866i)16-s + (−1.22 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + 6-s + (−1.22 − 0.707i)7-s + (0.707 + 0.707i)8-s + 10-s + (1 + i)11-s + (0.965 + 0.258i)12-s + (0.258 + 0.965i)13-s + (−0.999 − i)14-s + (0.866 − 0.499i)15-s + (0.500 + 0.866i)16-s + (−1.22 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.219089349\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.219089349\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 37 | \( 1 + (0.965 - 0.258i)T \) |
good | 3 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.000363429623346538915892409915, −8.250111044914577416466102962706, −6.96494305164395984665273427364, −6.68790440974132857642043205056, −6.26847176308830703708525223118, −4.85847493538260941558489579858, −4.24989490136653650199081294486, −3.42590095139242383781322223446, −2.35229586414907195073418928406, −1.87080098186009620701592786835,
1.64905434566287743034534347852, 2.65800203390311993806934389188, 3.37372089916891360089485914056, 3.64203260469956747632775179996, 5.16808085924188456514195984792, 5.93692471231811940268287490446, 6.29016949455628464090618409392, 7.11229791026891157280496313906, 8.387380938418681074760457323749, 9.040547132685159259733299452788