| L(s) = 1 | + 9·5-s + 7-s − 63·11-s + 28·13-s − 72·17-s + 98·19-s + 126·23-s − 44·25-s − 126·29-s + 259·31-s + 9·35-s − 386·37-s + 450·41-s − 34·43-s − 54·47-s − 342·49-s − 693·53-s − 567·55-s − 180·59-s + 280·61-s + 252·65-s − 586·67-s + 504·71-s + 161·73-s − 63·77-s − 440·79-s − 999·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 0.0539·7-s − 1.72·11-s + 0.597·13-s − 1.02·17-s + 1.18·19-s + 1.14·23-s − 0.351·25-s − 0.806·29-s + 1.50·31-s + 0.0434·35-s − 1.71·37-s + 1.71·41-s − 0.120·43-s − 0.167·47-s − 0.997·49-s − 1.79·53-s − 1.39·55-s − 0.397·59-s + 0.587·61-s + 0.480·65-s − 1.06·67-s + 0.842·71-s + 0.258·73-s − 0.0932·77-s − 0.626·79-s − 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 63 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 98 T + p^{3} T^{2} \) |
| 23 | \( 1 - 126 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 259 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 - 450 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 + 54 T + p^{3} T^{2} \) |
| 53 | \( 1 + 693 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 280 T + p^{3} T^{2} \) |
| 67 | \( 1 + 586 T + p^{3} T^{2} \) |
| 71 | \( 1 - 504 T + p^{3} T^{2} \) |
| 73 | \( 1 - 161 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 999 T + p^{3} T^{2} \) |
| 89 | \( 1 + 882 T + p^{3} T^{2} \) |
| 97 | \( 1 + 721 T + p^{3} T^{2} \) |
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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
−8.564365395505231964383296652844, −7.80415030506279252166298689395, −6.98325725312641300789382570786, −6.02720189637988636522029661335, −5.32790713097877843163236029638, −4.62082910644305500561267748450, −3.23647854704156799629265537357, −2.47189364341635543342581017837, −1.38486030468573446544176870267, 0,
1.38486030468573446544176870267, 2.47189364341635543342581017837, 3.23647854704156799629265537357, 4.62082910644305500561267748450, 5.32790713097877843163236029638, 6.02720189637988636522029661335, 6.98325725312641300789382570786, 7.80415030506279252166298689395, 8.564365395505231964383296652844
