| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s + 2·12-s − 5·13-s + 15-s − 4·16-s − 2·18-s + 19-s + 2·20-s + 25-s + 10·26-s + 27-s + 6·29-s − 2·30-s + 8·32-s + 2·36-s − 6·37-s − 2·38-s − 5·39-s + 2·41-s + 11·43-s + 45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s − 1.38·13-s + 0.258·15-s − 16-s − 0.471·18-s + 0.229·19-s + 0.447·20-s + 1/5·25-s + 1.96·26-s + 0.192·27-s + 1.11·29-s − 0.365·30-s + 1.41·32-s + 1/3·36-s − 0.986·37-s − 0.324·38-s − 0.800·39-s + 0.312·41-s + 1.67·43-s + 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.07571121369146, −13.79347418620507, −13.21783799569617, −12.39722359839428, −12.28768091970786, −11.54369370335725, −10.87630049760311, −10.42615465690878, −10.04007438748254, −9.583675663884451, −9.128717465442693, −8.743990214835250, −8.194781356507611, −7.543498074852471, −7.338372002152816, −6.765227221686120, −6.097563161060245, −5.417891165055970, −4.620318056968146, −4.397997431751128, −3.318493345028223, −2.696800327108667, −2.220489257448306, −1.561295767330892, −0.8599936908968200, 0,
0.8599936908968200, 1.561295767330892, 2.220489257448306, 2.696800327108667, 3.318493345028223, 4.397997431751128, 4.620318056968146, 5.417891165055970, 6.097563161060245, 6.765227221686120, 7.338372002152816, 7.543498074852471, 8.194781356507611, 8.743990214835250, 9.128717465442693, 9.583675663884451, 10.04007438748254, 10.42615465690878, 10.87630049760311, 11.54369370335725, 12.28768091970786, 12.39722359839428, 13.21783799569617, 13.79347418620507, 14.07571121369146
