| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s − 11-s − 12-s − 6·13-s + 3·15-s + 16-s + 5·17-s − 18-s − 6·19-s − 3·20-s + 22-s + 5·23-s + 24-s + 4·25-s + 6·26-s − 27-s − 6·29-s − 3·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.774·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s − 0.670·20-s + 0.213·22-s + 1.04·23-s + 0.204·24-s + 4/5·25-s + 1.17·26-s − 0.192·27-s − 1.11·29-s − 0.547·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2850366309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2850366309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.437156545038805140814932730332, −7.957526633191908617876120023128, −7.12910487439486860495499701538, −6.86136903500212564394979157694, −5.51605635968754426443992710401, −4.92689675195570829518821943240, −3.92692428639608188414387845801, −3.05331510142960887346457349810, −1.84523390935343279940589930817, −0.35274093810809910411564142375,
0.35274093810809910411564142375, 1.84523390935343279940589930817, 3.05331510142960887346457349810, 3.92692428639608188414387845801, 4.92689675195570829518821943240, 5.51605635968754426443992710401, 6.86136903500212564394979157694, 7.12910487439486860495499701538, 7.957526633191908617876120023128, 8.437156545038805140814932730332
