| L(s) = 1 | − 3·3-s − 18·5-s + 9·9-s + 72·11-s + 34·13-s + 54·15-s − 6·17-s + 92·19-s + 180·23-s + 199·25-s − 27·27-s − 114·29-s + 56·31-s − 216·33-s − 34·37-s − 102·39-s − 6·41-s − 164·43-s − 162·45-s + 168·47-s + 18·51-s + 654·53-s − 1.29e3·55-s − 276·57-s − 492·59-s + 250·61-s − 612·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.60·5-s + 1/3·9-s + 1.97·11-s + 0.725·13-s + 0.929·15-s − 0.0856·17-s + 1.11·19-s + 1.63·23-s + 1.59·25-s − 0.192·27-s − 0.729·29-s + 0.324·31-s − 1.13·33-s − 0.151·37-s − 0.418·39-s − 0.0228·41-s − 0.581·43-s − 0.536·45-s + 0.521·47-s + 0.0494·51-s + 1.69·53-s − 3.17·55-s − 0.641·57-s − 1.08·59-s + 0.524·61-s − 1.16·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.665272423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665272423\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 114 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 + 56 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.812948539511748379741124410690, −7.69840983000855273992389626882, −7.07812082220494322439131108968, −6.49655066184860625460549686397, −5.45324231427672475496783164110, −4.47071091133701093286299187459, −3.82752079659192344412890984796, −3.19649750944373798321339946267, −1.35480510145958541982131211275, −0.68499578352339498556605518934,
0.68499578352339498556605518934, 1.35480510145958541982131211275, 3.19649750944373798321339946267, 3.82752079659192344412890984796, 4.47071091133701093286299187459, 5.45324231427672475496783164110, 6.49655066184860625460549686397, 7.07812082220494322439131108968, 7.69840983000855273992389626882, 8.812948539511748379741124410690
