| L(s) = 1 | + 0.381·2-s + 2·3-s − 1.85·4-s − 2.61·5-s + 0.763·6-s − 1.47·8-s + 9-s − 10-s − 0.381·11-s − 3.70·12-s + 6.85·13-s − 5.23·15-s + 3.14·16-s − 2.23·17-s + 0.381·18-s − 5.70·19-s + 4.85·20-s − 0.145·22-s − 4.09·23-s − 2.94·24-s + 1.85·25-s + 2.61·26-s − 4·27-s + 8.23·29-s − 2·30-s + 5·31-s + 4.14·32-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 1.15·3-s − 0.927·4-s − 1.17·5-s + 0.311·6-s − 0.520·8-s + 0.333·9-s − 0.316·10-s − 0.115·11-s − 1.07·12-s + 1.90·13-s − 1.35·15-s + 0.786·16-s − 0.542·17-s + 0.0900·18-s − 1.30·19-s + 1.08·20-s − 0.0311·22-s − 0.852·23-s − 0.600·24-s + 0.370·25-s + 0.513·26-s − 0.769·27-s + 1.52·29-s − 0.365·30-s + 0.898·31-s + 0.732·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743931238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743931238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 0.381T + 11T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.854T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.103994254184039786055625603627, −7.85042420016822690290763161422, −6.49287637619021865420002270962, −6.09472861255790417308918784009, −4.80117318033936074862025077451, −4.24081360227491438669514378552, −3.61519583927091551493717342356, −3.16508730197638154685819036851, −1.98985052275062812249766077202, −0.61994808537038113971831207402,
0.61994808537038113971831207402, 1.98985052275062812249766077202, 3.16508730197638154685819036851, 3.61519583927091551493717342356, 4.24081360227491438669514378552, 4.80117318033936074862025077451, 6.09472861255790417308918784009, 6.49287637619021865420002270962, 7.85042420016822690290763161422, 8.103994254184039786055625603627
