| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s + 2·13-s + 15-s + 2·17-s + 21-s + 8·23-s + 25-s + 27-s + 2·29-s − 8·31-s − 33-s + 35-s + 6·37-s + 2·39-s − 2·41-s + 12·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 10·53-s − 55-s − 12·59-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 + 0.485·17-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.312·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.134·55-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.410945437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.410945437\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.60353001507443332445330998222, −7.28389479212378605856560039519, −6.29379263370521105790022028163, −5.68358354305395598636222034241, −4.93165614238949853550990691782, −4.22527027767751750312832091883, −3.29414460265451037611846848967, −2.69406055199666445829374404466, −1.74366384728526952317804940208, −0.911591052733787401722674452379,
0.911591052733787401722674452379, 1.74366384728526952317804940208, 2.69406055199666445829374404466, 3.29414460265451037611846848967, 4.22527027767751750312832091883, 4.93165614238949853550990691782, 5.68358354305395598636222034241, 6.29379263370521105790022028163, 7.28389479212378605856560039519, 7.60353001507443332445330998222
