| L(s) = 1 | − 2·5-s − 2·13-s + 2·17-s − 19-s − 25-s + 2·29-s + 4·31-s + 2·37-s + 6·41-s − 4·43-s − 7·49-s − 10·53-s − 4·59-s + 2·61-s + 4·65-s + 12·67-s + 6·73-s − 4·79-s + 8·83-s − 4·85-s − 6·89-s + 2·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s + 0.485·17-s − 0.229·19-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 49-s − 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.496·65-s + 1.46·67-s + 0.702·73-s − 0.450·79-s + 0.878·83-s − 0.433·85-s − 0.635·89-s + 0.205·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.70989347277828, −12.27800349932083, −12.05051186962612, −11.29559067292387, −11.23500793641096, −10.63230541272882, −9.989616727200671, −9.673033394918667, −9.293214542098336, −8.521743750520664, −8.105710851924074, −7.918772967129598, −7.293211730359651, −6.872359919623883, −6.298705444477573, −5.883301108200336, −5.156814861052948, −4.762938970586434, −4.281160383857387, −3.739979707780649, −3.226557273428677, −2.706478429083061, −2.082626053779138, −1.367086017065960, −0.6647169675869457, 0,
0.6647169675869457, 1.367086017065960, 2.082626053779138, 2.706478429083061, 3.226557273428677, 3.739979707780649, 4.281160383857387, 4.762938970586434, 5.156814861052948, 5.883301108200336, 6.298705444477573, 6.872359919623883, 7.293211730359651, 7.918772967129598, 8.105710851924074, 8.521743750520664, 9.293214542098336, 9.673033394918667, 9.989616727200671, 10.63230541272882, 11.23500793641096, 11.29559067292387, 12.05051186962612, 12.27800349932083, 12.70989347277828
