| L(s) = 1 | − 3-s − 3·5-s + 3·7-s + 9-s − 6·13-s + 3·15-s − 2·17-s + 2·19-s − 3·21-s + 6·23-s + 4·25-s − 27-s + 4·29-s − 5·31-s − 9·35-s − 37-s + 6·39-s + 2·41-s + 4·43-s − 3·45-s + 9·47-s + 2·49-s + 2·51-s + 3·53-s − 2·57-s − 7·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.13·7-s + 1/3·9-s − 1.66·13-s + 0.774·15-s − 0.485·17-s + 0.458·19-s − 0.654·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.742·29-s − 0.898·31-s − 1.52·35-s − 0.164·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.447·45-s + 1.31·47-s + 2/7·49-s + 0.280·51-s + 0.412·53-s − 0.264·57-s − 0.911·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.42547718934921515894861532578, −7.16048252696028348545347808537, −6.07609638984929845842757135009, −5.06877253382905111438026717918, −4.77506119219876845315651106056, −4.15122584995366202290148635405, −3.14782236812107632025249477478, −2.21854500407087355077468791406, −1.04610723379640327787234669641, 0,
1.04610723379640327787234669641, 2.21854500407087355077468791406, 3.14782236812107632025249477478, 4.15122584995366202290148635405, 4.77506119219876845315651106056, 5.06877253382905111438026717918, 6.07609638984929845842757135009, 7.16048252696028348545347808537, 7.42547718934921515894861532578
