| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 6·13-s + 15-s − 2·17-s − 4·19-s + 4·21-s + 8·23-s + 25-s − 27-s − 6·29-s + 4·35-s − 6·37-s + 6·39-s + 10·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s + 2·51-s + 10·53-s + 4·57-s + 6·61-s − 4·63-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s − 0.986·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.768·61-s − 0.503·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−11.70834245676778195792945574431, −10.69580346976699843945117017007, −9.770206916241007652686400963668, −8.931256392945798101264819577475, −7.31616880556813656693675020931, −6.73102949270507316916720435534, −5.45298154060946963380399619268, −4.19366402654330570965762069528, −2.75290686228331533750926083473, 0,
2.75290686228331533750926083473, 4.19366402654330570965762069528, 5.45298154060946963380399619268, 6.73102949270507316916720435534, 7.31616880556813656693675020931, 8.931256392945798101264819577475, 9.770206916241007652686400963668, 10.69580346976699843945117017007, 11.70834245676778195792945574431
