L(s) = 1 | − 2·3-s − 7-s + 9-s − 6·17-s + 6·19-s + 2·21-s − 23-s − 5·25-s + 4·27-s + 6·29-s − 8·31-s − 2·37-s − 2·41-s − 8·43-s + 8·47-s + 49-s + 12·51-s + 2·53-s − 12·57-s − 6·59-s − 63-s − 12·67-s + 2·69-s + 8·71-s − 6·73-s + 10·75-s + 16·79-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.274·53-s − 1.58·57-s − 0.781·59-s − 0.125·63-s − 1.46·67-s + 0.240·69-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6381734044\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6381734044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.46753370536440, −16.21714782236016, −15.55401276900372, −15.11924706818992, −14.15987317733247, −13.65109017999799, −13.18379534965632, −12.32770426383971, −11.90427594103538, −11.49169691514433, −10.69410055831462, −10.43427769286864, −9.486488263686222, −9.082098351540372, −8.235610494898363, −7.465957842846244, −6.759075048254671, −6.315964658007927, −5.545757619190799, −5.092190274657899, −4.282952096898660, −3.477681673587700, −2.571535099520529, −1.547729596637332, −0.4021030751198909,
0.4021030751198909, 1.547729596637332, 2.571535099520529, 3.477681673587700, 4.282952096898660, 5.092190274657899, 5.545757619190799, 6.315964658007927, 6.759075048254671, 7.465957842846244, 8.235610494898363, 9.082098351540372, 9.486488263686222, 10.43427769286864, 10.69410055831462, 11.49169691514433, 11.90427594103538, 12.32770426383971, 13.18379534965632, 13.65109017999799, 14.15987317733247, 15.11924706818992, 15.55401276900372, 16.21714782236016, 16.46753370536440