| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 2·13-s + 2·15-s − 4·17-s + 6·19-s − 25-s − 27-s − 8·29-s + 8·31-s − 33-s + 10·37-s − 2·39-s − 8·41-s − 2·43-s − 2·45-s + 8·47-s + 4·51-s − 2·53-s − 2·55-s − 6·57-s − 12·59-s − 10·61-s − 4·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.970·17-s + 1.37·19-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.174·33-s + 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.304·43-s − 0.298·45-s + 1.16·47-s + 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.794·57-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.149102567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.149102567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.82221538227275507734605765990, −7.42846124160177781591463423574, −6.54297286809666463797796948548, −5.98940884431177188957039743782, −5.11671587860455197842454084088, −4.36167357137437320039390160698, −3.75822578997708969858043174020, −2.89008687447053373852273904799, −1.64252900344152225108111960615, −0.58713146109383379384422071373,
0.58713146109383379384422071373, 1.64252900344152225108111960615, 2.89008687447053373852273904799, 3.75822578997708969858043174020, 4.36167357137437320039390160698, 5.11671587860455197842454084088, 5.98940884431177188957039743782, 6.54297286809666463797796948548, 7.42846124160177781591463423574, 7.82221538227275507734605765990
