| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s − 4·13-s − 15-s − 17-s − 8·19-s − 21-s − 23-s + 25-s − 27-s + 29-s − 9·31-s + 33-s + 35-s + 2·37-s + 4·39-s + 8·41-s + 43-s + 45-s + 8·47-s − 6·49-s + 51-s − 8·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.242·17-s − 1.83·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 1.61·31-s + 0.174·33-s + 0.169·35-s + 0.328·37-s + 0.640·39-s + 1.24·41-s + 0.152·43-s + 0.149·45-s + 1.16·47-s − 6/7·49-s + 0.140·51-s − 1.09·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.22647676616099, −12.83366556543394, −12.49418248646179, −12.15745687038760, −11.28754982344720, −11.10867854324028, −10.62817446553507, −10.17042045237229, −9.628428031251041, −9.211620977431787, −8.650806468839680, −8.089740952043428, −7.611888605371481, −7.007276057000042, −6.694475541538004, −5.867589101398229, −5.758313903383441, −5.006878439054243, −4.541571555053378, −4.165829644810461, −3.413352309270486, −2.524841676918405, −2.178257947954024, −1.645144083446430, −0.6735293765726212, 0,
0.6735293765726212, 1.645144083446430, 2.178257947954024, 2.524841676918405, 3.413352309270486, 4.165829644810461, 4.541571555053378, 5.006878439054243, 5.758313903383441, 5.867589101398229, 6.694475541538004, 7.007276057000042, 7.611888605371481, 8.089740952043428, 8.650806468839680, 9.211620977431787, 9.628428031251041, 10.17042045237229, 10.62817446553507, 11.10867854324028, 11.28754982344720, 12.15745687038760, 12.49418248646179, 12.83366556543394, 13.22647676616099
