| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 6·11-s − 4·13-s + 16-s + 2·19-s + 20-s + 6·22-s + 3·23-s + 25-s − 4·26-s + 3·29-s + 8·31-s + 32-s − 4·37-s + 2·38-s + 40-s − 9·41-s − 7·43-s + 6·44-s + 3·46-s + 50-s − 4·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.80·11-s − 1.10·13-s + 1/4·16-s + 0.458·19-s + 0.223·20-s + 1.27·22-s + 0.625·23-s + 1/5·25-s − 0.784·26-s + 0.557·29-s + 1.43·31-s + 0.176·32-s − 0.657·37-s + 0.324·38-s + 0.158·40-s − 1.40·41-s − 1.06·43-s + 0.904·44-s + 0.442·46-s + 0.141·50-s − 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.832178025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.832178025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.446057847965057961234874071223, −7.33627374250111876505552017455, −6.74180059874853741828438330264, −6.28329671170162028085026581941, −5.23564994296840140136361153527, −4.72969970959816709399868353364, −3.81237566315548845936777504653, −3.02727210544214318041323551049, −2.03834482847849339410432158845, −1.05594984404650199095603752569,
1.05594984404650199095603752569, 2.03834482847849339410432158845, 3.02727210544214318041323551049, 3.81237566315548845936777504653, 4.72969970959816709399868353364, 5.23564994296840140136361153527, 6.28329671170162028085026581941, 6.74180059874853741828438330264, 7.33627374250111876505552017455, 8.446057847965057961234874071223
