| L(s) = 1 | − 3·3-s − 5·5-s + 8·7-s + 9·9-s − 20·11-s − 22·13-s + 15·15-s − 14·17-s − 76·19-s − 24·21-s + 56·23-s + 25·25-s − 27·27-s + 154·29-s + 160·31-s + 60·33-s − 40·35-s + 162·37-s + 66·39-s − 390·41-s − 388·43-s − 45·45-s − 544·47-s − 279·49-s + 42·51-s + 210·53-s + 100·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.431·7-s + 1/3·9-s − 0.548·11-s − 0.469·13-s + 0.258·15-s − 0.199·17-s − 0.917·19-s − 0.249·21-s + 0.507·23-s + 1/5·25-s − 0.192·27-s + 0.986·29-s + 0.926·31-s + 0.316·33-s − 0.193·35-s + 0.719·37-s + 0.270·39-s − 1.48·41-s − 1.37·43-s − 0.149·45-s − 1.68·47-s − 0.813·49-s + 0.115·51-s + 0.544·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.182238831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182238831\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 154 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 544 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 794 T + p^{3} T^{2} \) |
| 67 | \( 1 - 148 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 858 T + p^{3} T^{2} \) |
| 79 | \( 1 - 144 T + p^{3} T^{2} \) |
| 83 | \( 1 + 316 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 97 | \( 1 - 994 T + p^{3} T^{2} \) |
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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
−9.920370770622587982093679468834, −8.577104988638664599862072123888, −8.078582706357201966590393931600, −6.99657322751975767908993439152, −6.32689750948624946516389548489, −5.04415091672794830099158523216, −4.61974270611407053025018143705, −3.30029145710657066623189560582, −2.03425927442763433429075232420, −0.59137124545077071954444478700,
0.59137124545077071954444478700, 2.03425927442763433429075232420, 3.30029145710657066623189560582, 4.61974270611407053025018143705, 5.04415091672794830099158523216, 6.32689750948624946516389548489, 6.99657322751975767908993439152, 8.078582706357201966590393931600, 8.577104988638664599862072123888, 9.920370770622587982093679468834
