| L(s) = 1 | + 3·3-s + 2·5-s + 7·7-s + 9·9-s − 12·11-s + 66·13-s + 6·15-s − 70·17-s + 92·19-s + 21·21-s + 16·23-s − 121·25-s + 27·27-s + 122·29-s + 64·31-s − 36·33-s + 14·35-s + 306·37-s + 198·39-s + 50·41-s − 20·43-s + 18·45-s − 176·47-s + 49·49-s − 210·51-s − 526·53-s − 24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.178·5-s + 0.377·7-s + 1/3·9-s − 0.328·11-s + 1.40·13-s + 0.103·15-s − 0.998·17-s + 1.11·19-s + 0.218·21-s + 0.145·23-s − 0.967·25-s + 0.192·27-s + 0.781·29-s + 0.370·31-s − 0.189·33-s + 0.0676·35-s + 1.35·37-s + 0.812·39-s + 0.190·41-s − 0.0709·43-s + 0.0596·45-s − 0.546·47-s + 1/7·49-s − 0.576·51-s − 1.36·53-s − 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.210186220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.210186220\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 64 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 - 50 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 176 T + p^{3} T^{2} \) |
| 53 | \( 1 + 526 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 818 T + p^{3} T^{2} \) |
| 67 | \( 1 - 228 T + p^{3} T^{2} \) |
| 71 | \( 1 - 864 T + p^{3} T^{2} \) |
| 73 | \( 1 - 106 T + p^{3} T^{2} \) |
| 79 | \( 1 - 736 T + p^{3} T^{2} \) |
| 83 | \( 1 - 588 T + p^{3} T^{2} \) |
| 89 | \( 1 - 146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1214 T + p^{3} T^{2} \) |
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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
−9.265762067207832598505939916627, −8.296571092084978213255651826269, −7.903058910248512738659979835825, −6.76638812625779447837463301935, −6.00208631045043516646634987299, −4.95451418242660377194501243276, −4.02204930138477824936139148830, −3.06815959489347121624332285621, −2.00239356583418777491801239594, −0.908658154776289415444972547010,
0.908658154776289415444972547010, 2.00239356583418777491801239594, 3.06815959489347121624332285621, 4.02204930138477824936139148830, 4.95451418242660377194501243276, 6.00208631045043516646634987299, 6.76638812625779447837463301935, 7.903058910248512738659979835825, 8.296571092084978213255651826269, 9.265762067207832598505939916627
