| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s − 2·9-s − 12-s − 2·13-s − 16-s − 2·17-s − 2·18-s − 6·19-s − 3·23-s − 3·24-s − 2·26-s − 5·27-s + 7·29-s − 2·31-s + 5·32-s − 2·34-s + 2·36-s − 8·37-s − 6·38-s − 2·39-s − 5·41-s + 7·43-s − 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s − 2/3·9-s − 0.288·12-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 1.37·19-s − 0.625·23-s − 0.612·24-s − 0.392·26-s − 0.962·27-s + 1.29·29-s − 0.359·31-s + 0.883·32-s − 0.342·34-s + 1/3·36-s − 1.31·37-s − 0.973·38-s − 0.320·39-s − 0.780·41-s + 1.06·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.055060181444677902680642741683, −8.663910471398103803302152473134, −7.84358414802516235850066127888, −6.63199518238891215109968686447, −5.85074777908685625413668604665, −4.88474901346517409165234199202, −4.10696249494892867437330846726, −3.14124732040647939933466919756, −2.20278616257192938769552788629, 0,
2.20278616257192938769552788629, 3.14124732040647939933466919756, 4.10696249494892867437330846726, 4.88474901346517409165234199202, 5.85074777908685625413668604665, 6.63199518238891215109968686447, 7.84358414802516235850066127888, 8.663910471398103803302152473134, 9.055060181444677902680642741683
