| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 2·13-s + 2·15-s + 6·17-s − 19-s − 21-s − 25-s − 27-s − 10·29-s − 4·31-s − 2·35-s + 2·37-s + 2·39-s + 10·41-s + 4·43-s − 2·45-s + 12·47-s + 49-s − 6·51-s − 2·53-s + 57-s − 4·59-s − 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.274·53-s + 0.132·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.944422877285182503858228499197, −7.65946887673850072402659705371, −6.98646490113563992100765325439, −5.75167870284283557472878681421, −5.42948539521435568164584301677, −4.26033783633881799492746129650, −3.79552704157897294815360667984, −2.57621638970045974455500982202, −1.29130635360213814076416690073, 0,
1.29130635360213814076416690073, 2.57621638970045974455500982202, 3.79552704157897294815360667984, 4.26033783633881799492746129650, 5.42948539521435568164584301677, 5.75167870284283557472878681421, 6.98646490113563992100765325439, 7.65946887673850072402659705371, 7.944422877285182503858228499197
