| L(s) = 1 | + 0.561·2-s − 7.68·4-s + 29.8·7-s − 8.80·8-s + 56.0·11-s − 20.4·13-s + 16.7·14-s + 56.5·16-s − 18.5·17-s − 70.7·19-s + 31.4·22-s − 111.·23-s − 11.4·26-s − 229.·28-s + 72.7·29-s + 106.·31-s + 102.·32-s − 10.4·34-s − 100.·37-s − 39.7·38-s − 307.·41-s + 479.·43-s − 430.·44-s − 62.8·46-s + 472.·47-s + 547.·49-s + 156.·52-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.960·4-s + 1.61·7-s − 0.389·8-s + 1.53·11-s − 0.435·13-s + 0.319·14-s + 0.883·16-s − 0.264·17-s − 0.854·19-s + 0.304·22-s − 1.01·23-s − 0.0865·26-s − 1.54·28-s + 0.465·29-s + 0.615·31-s + 0.564·32-s − 0.0525·34-s − 0.446·37-s − 0.169·38-s − 1.17·41-s + 1.69·43-s − 1.47·44-s − 0.201·46-s + 1.46·47-s + 1.59·49-s + 0.418·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.196875604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.196875604\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.561T + 8T^{2} \) |
| 7 | \( 1 - 29.8T + 343T^{2} \) |
| 11 | \( 1 - 56.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 479.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 429.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 443.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 465.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 234.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 275.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 309.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
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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
−10.08022329848681271509008044625, −9.018678739746675544522114589089, −8.528123847785564638905701147626, −7.64937875811913948143627646711, −6.44517738093550751934328982946, −5.37934042130951759098901305763, −4.43738167060697408216382598109, −3.92896462261293155915115610912, −2.10895229183235512015408852436, −0.888135214714639769298303840061,
0.888135214714639769298303840061, 2.10895229183235512015408852436, 3.92896462261293155915115610912, 4.43738167060697408216382598109, 5.37934042130951759098901305763, 6.44517738093550751934328982946, 7.64937875811913948143627646711, 8.528123847785564638905701147626, 9.018678739746675544522114589089, 10.08022329848681271509008044625
