| L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 4·17-s + 2·19-s − 4·23-s − 5·25-s + 10·29-s − 6·31-s − 37-s − 6·41-s + 4·43-s − 12·47-s + 49-s − 6·53-s − 2·59-s + 12·61-s + 3·63-s − 12·67-s − 8·71-s + 2·73-s − 4·77-s − 12·79-s + 9·81-s − 16·83-s + 12·89-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 0.970·17-s + 0.458·19-s − 0.834·23-s − 25-s + 1.85·29-s − 1.07·31-s − 0.164·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.260·59-s + 1.53·61-s + 0.377·63-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.455·77-s − 1.35·79-s + 81-s − 1.75·83-s + 1.27·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.721234168224049329162571849761, −8.136829150281921097219581170343, −7.01860491936496261393052940157, −6.36099873001269398888741200825, −5.69607433952137350924404521939, −4.60174461162217279988964528524, −3.72225411159995483231744620868, −2.82309166704277733457457789789, −1.63242598314451731274866577199, 0,
1.63242598314451731274866577199, 2.82309166704277733457457789789, 3.72225411159995483231744620868, 4.60174461162217279988964528524, 5.69607433952137350924404521939, 6.36099873001269398888741200825, 7.01860491936496261393052940157, 8.136829150281921097219581170343, 8.721234168224049329162571849761
