| L(s) = 1 | + 3·3-s − 5-s + 6·9-s − 6·13-s − 3·15-s + 4·17-s − 6·19-s − 3·23-s − 4·25-s + 9·27-s − 4·29-s + 9·31-s − 7·37-s − 18·39-s + 2·41-s − 6·43-s − 6·45-s − 12·47-s − 7·49-s + 12·51-s − 2·53-s − 18·57-s + 9·59-s + 8·61-s + 6·65-s − 15·67-s − 9·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s − 1.66·13-s − 0.774·15-s + 0.970·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s + 1.73·27-s − 0.742·29-s + 1.61·31-s − 1.15·37-s − 2.88·39-s + 0.312·41-s − 0.914·43-s − 0.894·45-s − 1.75·47-s − 49-s + 1.68·51-s − 0.274·53-s − 2.38·57-s + 1.17·59-s + 1.02·61-s + 0.744·65-s − 1.83·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.77504752207978386728366476611, −7.08072343183803009576323277610, −6.37305639235248964419573519168, −5.18502090217935779773115652171, −4.46713554310869373480031428571, −3.77327816150142389344477767412, −3.09410655009033832646896370919, −2.32539615240176503621700363929, −1.67933822551948351378331307814, 0,
1.67933822551948351378331307814, 2.32539615240176503621700363929, 3.09410655009033832646896370919, 3.77327816150142389344477767412, 4.46713554310869373480031428571, 5.18502090217935779773115652171, 6.37305639235248964419573519168, 7.08072343183803009576323277610, 7.77504752207978386728366476611
