| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 2·13-s − 15-s + 2·17-s + 19-s + 4·21-s + 25-s − 27-s − 6·29-s − 8·31-s − 4·35-s + 2·37-s − 2·39-s + 6·41-s + 8·43-s + 45-s + 8·47-s + 9·49-s − 2·51-s − 10·53-s − 57-s − 12·59-s − 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s − 0.132·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.74239971493544785078819208111, −7.19228762811754192186951882164, −6.28170854953327770279618565314, −5.93092761786036822402412864098, −5.24159634231935644354706530697, −4.06179769154884615194864576013, −3.42921025915202889612694406225, −2.47741598129931841696906268390, −1.24347041780174455905697637861, 0,
1.24347041780174455905697637861, 2.47741598129931841696906268390, 3.42921025915202889612694406225, 4.06179769154884615194864576013, 5.24159634231935644354706530697, 5.93092761786036822402412864098, 6.28170854953327770279618565314, 7.19228762811754192186951882164, 7.74239971493544785078819208111
