| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 2·13-s − 15-s − 6·17-s + 8·19-s + 21-s − 6·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 33-s − 35-s + 2·37-s + 2·39-s + 8·43-s − 45-s − 12·47-s + 49-s − 6·51-s + 6·53-s + 55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s + 1.21·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.320555304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.320555304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.239827373992631445973936664099, −7.77021842427155557820166533717, −6.97477756558267793843872697439, −6.23425244740462665236858274294, −5.26241214785282146370725796022, −4.48796719388501819451072526975, −3.77784555461614582637697828436, −2.89859414506902250350137509125, −2.02918532336666136868637133674, −0.835183511952370404605183958765,
0.835183511952370404605183958765, 2.02918532336666136868637133674, 2.89859414506902250350137509125, 3.77784555461614582637697828436, 4.48796719388501819451072526975, 5.26241214785282146370725796022, 6.23425244740462665236858274294, 6.97477756558267793843872697439, 7.77021842427155557820166533717, 8.239827373992631445973936664099
