| L(s) = 1 | − 2.73·5-s + 7-s − 2.24·11-s − 0.458·13-s − 7.44·17-s + 4.24·19-s − 4.70·23-s + 2.50·25-s − 1.73·29-s − 1.73·31-s − 2.73·35-s + 2.49·37-s − 4.73·41-s − 1.96·43-s + 9.90·47-s + 49-s + 5.42·53-s + 6.15·55-s + 6.47·59-s + 5.23·61-s + 1.25·65-s − 2.77·67-s + 0.703·71-s − 7.96·73-s − 2.24·77-s − 6.66·79-s + 3.50·83-s + ⋯ |
| L(s) = 1 | − 1.22·5-s + 0.377·7-s − 0.676·11-s − 0.127·13-s − 1.80·17-s + 0.973·19-s − 0.980·23-s + 0.501·25-s − 0.323·29-s − 0.312·31-s − 0.463·35-s + 0.410·37-s − 0.740·41-s − 0.299·43-s + 1.44·47-s + 0.142·49-s + 0.744·53-s + 0.829·55-s + 0.843·59-s + 0.670·61-s + 0.155·65-s − 0.339·67-s + 0.0835·71-s − 0.932·73-s − 0.255·77-s − 0.750·79-s + 0.384·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9039766022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9039766022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 0.458T + 13T^{2} \) |
| 17 | \( 1 + 7.44T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 2.77T + 67T^{2} \) |
| 71 | \( 1 - 0.703T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 + 6.66T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.055940450473674886832702337750, −7.39503012121939899724682770947, −6.91436579278990664140837084522, −5.87560489995100177750862077293, −5.11715136489752782023161574461, −4.31570696992215995681891220499, −3.82608333537007061913668484015, −2.78951034116507858121468846128, −1.92889648426033721430511828440, −0.47982309671297790410679821517,
0.47982309671297790410679821517, 1.92889648426033721430511828440, 2.78951034116507858121468846128, 3.82608333537007061913668484015, 4.31570696992215995681891220499, 5.11715136489752782023161574461, 5.87560489995100177750862077293, 6.91436579278990664140837084522, 7.39503012121939899724682770947, 8.055940450473674886832702337750
