| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 13-s + 15-s + 6·17-s + 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 4·35-s + 2·37-s + 39-s + 10·41-s − 4·43-s − 45-s − 8·47-s + 9·49-s − 6·51-s + 2·53-s − 4·55-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.413869633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413869633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.26726575159374, −16.02069059962189, −15.20625945127886, −14.72021620563162, −14.06276021443453, −13.40662631091681, −12.77135093183094, −12.19729580084370, −11.94899691698993, −11.28822569958012, −10.42514679624063, −9.994253964361173, −9.443470278459418, −8.894849178152757, −8.004962797876854, −7.358567276432738, −6.605463044733187, −6.339727291221506, −5.624530264137823, −4.700869537124198, −4.105026771035838, −3.224013769789623, −2.861762228471790, −1.325314823724982, −0.6213120713653383,
0.6213120713653383, 1.325314823724982, 2.861762228471790, 3.224013769789623, 4.105026771035838, 4.700869537124198, 5.624530264137823, 6.339727291221506, 6.605463044733187, 7.358567276432738, 8.004962797876854, 8.894849178152757, 9.443470278459418, 9.994253964361173, 10.42514679624063, 11.28822569958012, 11.94899691698993, 12.19729580084370, 12.77135093183094, 13.40662631091681, 14.06276021443453, 14.72021620563162, 15.20625945127886, 16.02069059962189, 16.26726575159374
