L(s) = 1 | + 2·3-s + (−2 − i)5-s + 2i·7-s + 9-s + 4i·11-s − 4·13-s + (−4 − 2i)15-s + 4i·19-s + 4i·21-s + 2i·23-s + (3 + 4i)25-s − 4·27-s + 2i·29-s + 8i·33-s + (2 − 4i)35-s + ⋯ |
L(s) = 1 | + 1.15·3-s + (−0.894 − 0.447i)5-s + 0.755i·7-s + 0.333·9-s + 1.20i·11-s − 1.10·13-s + (−1.03 − 0.516i)15-s + 0.917i·19-s + 0.872i·21-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 0.769·27-s + 0.371i·29-s + 1.39i·33-s + (0.338 − 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.312430941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.312430941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.512218371182951573576753475225, −9.180494455103817805734306839866, −8.200925432549830308994829089880, −7.71642686350118672014726087530, −6.97003929060344258222343922688, −5.55337239052502009125804132707, −4.67939603112629430874142962033, −3.75965242751613963035974419456, −2.76404632955529745201695174867, −1.81354972869085902912592111415,
0.44969418673227156163199383703, 2.42079975713106507910109817716, 3.21124637443676574041708568434, 3.94880278033456316785246763742, 4.95464885666485690105618443325, 6.32625140809115171331283741203, 7.29988577985083121264977916064, 7.75712121214731564645963326270, 8.591262282401732317022432716569, 9.197084419764676463655204011125