| L(s) = 1 | + 0.414·2-s + 3-s − 1.82·4-s + 3.41·5-s + 0.414·6-s − 1.58·8-s + 9-s + 1.41·10-s − 2·11-s − 1.82·12-s + 2.58·13-s + 3.41·15-s + 3·16-s − 2.24·17-s + 0.414·18-s − 2.82·19-s − 6.24·20-s − 0.828·22-s − 7.65·23-s − 1.58·24-s + 6.65·25-s + 1.07·26-s + 27-s − 6.82·29-s + 1.41·30-s − 1.17·31-s + 4.41·32-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.577·3-s − 0.914·4-s + 1.52·5-s + 0.169·6-s − 0.560·8-s + 0.333·9-s + 0.447·10-s − 0.603·11-s − 0.527·12-s + 0.717·13-s + 0.881·15-s + 0.750·16-s − 0.543·17-s + 0.0976·18-s − 0.648·19-s − 1.39·20-s − 0.176·22-s − 1.59·23-s − 0.323·24-s + 1.33·25-s + 0.210·26-s + 0.192·27-s − 1.26·29-s + 0.258·30-s − 0.210·31-s + 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468652443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468652443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−13.27991901401338177920979096245, −12.58858414642191942982135369824, −10.77716644511339454537220883340, −9.783092558409999515652215080723, −9.076856079002227489979725189354, −8.059283027700205192169067404971, −6.28194973118784079134047204927, −5.35315698187524417229152126700, −3.90425253661183828324360261573, −2.16504759383512122024690547881,
2.16504759383512122024690547881, 3.90425253661183828324360261573, 5.35315698187524417229152126700, 6.28194973118784079134047204927, 8.059283027700205192169067404971, 9.076856079002227489979725189354, 9.783092558409999515652215080723, 10.77716644511339454537220883340, 12.58858414642191942982135369824, 13.27991901401338177920979096245
