| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 4·11-s − 6·13-s − 15-s + 2·17-s + 4·19-s − 4·21-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 4·35-s + 10·37-s + 6·39-s + 2·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 2·53-s + 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s + 1.64·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743125803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743125803\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−9.758345846761255680723166486652, −9.569887168096234669182184873626, −8.218385815535771240939930654544, −7.49235926303250200960974861538, −6.65509689812873112624861887690, −5.45857154341122253575185626745, −4.97527404703815748732112348741, −3.93952361309394612597846303015, −2.28452944523597307623702961432, −1.18879387693747021147526651062,
1.18879387693747021147526651062, 2.28452944523597307623702961432, 3.93952361309394612597846303015, 4.97527404703815748732112348741, 5.45857154341122253575185626745, 6.65509689812873112624861887690, 7.49235926303250200960974861538, 8.218385815535771240939930654544, 9.569887168096234669182184873626, 9.758345846761255680723166486652
