| L(s) = 1 | + 3-s − 2·5-s − 7-s − 2·9-s − 13-s − 2·15-s + 8·17-s + 2·19-s − 21-s − 25-s − 5·27-s + 7·29-s + 7·31-s + 2·35-s − 2·37-s − 39-s − 11·41-s − 4·43-s + 4·45-s + 3·47-s + 49-s + 8·51-s + 6·53-s + 2·57-s − 4·59-s + 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.277·13-s − 0.516·15-s + 1.94·17-s + 0.458·19-s − 0.218·21-s − 1/5·25-s − 0.962·27-s + 1.29·29-s + 1.25·31-s + 0.338·35-s − 0.328·37-s − 0.160·39-s − 1.71·41-s − 0.609·43-s + 0.596·45-s + 0.437·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 0.264·57-s − 0.520·59-s + 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.25943388135790, −12.50104883912502, −12.04117209775580, −11.86526948469346, −11.55908150412180, −10.73257875563363, −10.22828664443878, −9.902881309038357, −9.484501064982017, −8.711438828806627, −8.411151228915751, −8.047155205774737, −7.501861498455019, −7.202292512437665, −6.409902307259931, −6.060948906150825, −5.279271106711704, −5.045323975177187, −4.269374495428015, −3.635920126237069, −3.293084265825517, −2.902638158796900, −2.283762619428381, −1.390018353178326, −0.7665571417326861, 0,
0.7665571417326861, 1.390018353178326, 2.283762619428381, 2.902638158796900, 3.293084265825517, 3.635920126237069, 4.269374495428015, 5.045323975177187, 5.279271106711704, 6.060948906150825, 6.409902307259931, 7.202292512437665, 7.501861498455019, 8.047155205774737, 8.411151228915751, 8.711438828806627, 9.484501064982017, 9.902881309038357, 10.22828664443878, 10.73257875563363, 11.55908150412180, 11.86526948469346, 12.04117209775580, 12.50104883912502, 13.25943388135790
