| L(s) = 1 | − 2·2-s − 5·3-s + 4·4-s + 10·6-s + 7·7-s − 8·8-s − 2·9-s − 11-s − 20·12-s − 7·13-s − 14·14-s + 16·16-s + 51·17-s + 4·18-s + 30·19-s − 35·21-s + 2·22-s + 50·23-s + 40·24-s + 14·26-s + 145·27-s + 28·28-s + 79·29-s − 212·31-s − 32·32-s + 5·33-s − 102·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s + 1/2·4-s + 0.680·6-s + 0.377·7-s − 0.353·8-s − 0.0740·9-s − 0.0274·11-s − 0.481·12-s − 0.149·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.0523·18-s + 0.362·19-s − 0.363·21-s + 0.0193·22-s + 0.453·23-s + 0.340·24-s + 0.105·26-s + 1.03·27-s + 0.188·28-s + 0.505·29-s − 1.22·31-s − 0.176·32-s + 0.0263·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + T + p^{3} T^{2} \) |
| 13 | \( 1 + 7 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 50 T + p^{3} T^{2} \) |
| 29 | \( 1 - 79 T + p^{3} T^{2} \) |
| 31 | \( 1 + 212 T + p^{3} T^{2} \) |
| 37 | \( 1 - 190 T + p^{3} T^{2} \) |
| 41 | \( 1 + 308 T + p^{3} T^{2} \) |
| 43 | \( 1 + 422 T + p^{3} T^{2} \) |
| 47 | \( 1 + 121 T + p^{3} T^{2} \) |
| 53 | \( 1 + 664 T + p^{3} T^{2} \) |
| 59 | \( 1 - 628 T + p^{3} T^{2} \) |
| 61 | \( 1 + 684 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1056 T + p^{3} T^{2} \) |
| 71 | \( 1 - 744 T + p^{3} T^{2} \) |
| 73 | \( 1 + 726 T + p^{3} T^{2} \) |
| 79 | \( 1 + 407 T + p^{3} T^{2} \) |
| 83 | \( 1 + 644 T + p^{3} T^{2} \) |
| 89 | \( 1 + 880 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1351 T + p^{3} T^{2} \) |
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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
−10.64717820914573832839960021218, −9.811962621357456218470796043261, −8.733437095664809291330699907880, −7.78082186310042783018312399008, −6.76471689004945006563886794118, −5.74901643546130697212082532467, −4.85829535208697526551158510569, −3.12091545218447459570526801300, −1.41699632861798656182482112152, 0,
1.41699632861798656182482112152, 3.12091545218447459570526801300, 4.85829535208697526551158510569, 5.74901643546130697212082532467, 6.76471689004945006563886794118, 7.78082186310042783018312399008, 8.733437095664809291330699907880, 9.811962621357456218470796043261, 10.64717820914573832839960021218
