| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 6·11-s + 4·13-s − 15-s + 6·17-s + 4·21-s + 6·23-s + 25-s − 27-s − 2·29-s + 6·33-s − 4·35-s + 8·37-s − 4·39-s + 10·41-s + 4·43-s + 45-s − 2·47-s + 9·49-s − 6·51-s − 10·53-s − 6·55-s + 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.04·33-s − 0.676·35-s + 1.31·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 0.809·55-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113442616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113442616\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.63461067182386, −12.41618883979518, −11.44581472837055, −11.21224204930225, −10.63979222752738, −10.37832633265454, −9.813105108988373, −9.495572562780108, −9.135893160929019, −8.303360069757463, −7.995991305587286, −7.390184859278123, −6.996980057567834, −6.403970715542774, −5.819373126361624, −5.760484362513020, −5.212476774007190, −4.604751541020434, −3.911268104054036, −3.371660257670744, −2.822346214217542, −2.607140394470126, −1.633089583924195, −0.8543208462607781, −0.5135497313478805,
0.5135497313478805, 0.8543208462607781, 1.633089583924195, 2.607140394470126, 2.822346214217542, 3.371660257670744, 3.911268104054036, 4.604751541020434, 5.212476774007190, 5.760484362513020, 5.819373126361624, 6.403970715542774, 6.996980057567834, 7.390184859278123, 7.995991305587286, 8.303360069757463, 9.135893160929019, 9.495572562780108, 9.813105108988373, 10.37832633265454, 10.63979222752738, 11.21224204930225, 11.44581472837055, 12.41618883979518, 12.63461067182386
