| L(s) = 1 | + 1.73·3-s − 1.46·5-s − 3.12·7-s + 0.0212·9-s + 2.23·11-s + 2.49·13-s − 2.54·15-s + 17-s + 6.76·19-s − 5.42·21-s + 5.38·23-s − 2.86·25-s − 5.17·27-s − 7.96·29-s + 9.56·31-s + 3.88·33-s + 4.56·35-s − 8.81·37-s + 4.33·39-s − 4.92·41-s − 4.38·43-s − 0.0310·45-s + 4.75·47-s + 2.75·49-s + 1.73·51-s + 5.15·53-s − 3.26·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.653·5-s − 1.18·7-s + 0.00708·9-s + 0.673·11-s + 0.692·13-s − 0.655·15-s + 0.242·17-s + 1.55·19-s − 1.18·21-s + 1.12·23-s − 0.572·25-s − 0.996·27-s − 1.47·29-s + 1.71·31-s + 0.675·33-s + 0.771·35-s − 1.44·37-s + 0.694·39-s − 0.768·41-s − 0.668·43-s − 0.00462·45-s + 0.693·47-s + 0.393·49-s + 0.243·51-s + 0.707·53-s − 0.439·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192200866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192200866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 4.38T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.18T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 - 1.91T + 73T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.74933476291320637045089281646, −7.32568423922322384727477565754, −6.55075060585405252289067953967, −5.84800813506415810511524200298, −5.00226104593326732022884324216, −3.87391237016746033297592441416, −3.37244883387504050784226854101, −3.07624455060412516624680827858, −1.85619850194037990768777165692, −0.69336283365340305241058469425,
0.69336283365340305241058469425, 1.85619850194037990768777165692, 3.07624455060412516624680827858, 3.37244883387504050784226854101, 3.87391237016746033297592441416, 5.00226104593326732022884324216, 5.84800813506415810511524200298, 6.55075060585405252289067953967, 7.32568423922322384727477565754, 7.74933476291320637045089281646
