| L(s) = 1 | + 2-s + 1.49·3-s + 4-s + 1.49·6-s + 1.01·7-s + 8-s − 0.763·9-s − 2.42·11-s + 1.49·12-s + 1.01·14-s + 16-s − 5.04·17-s − 0.763·18-s − 7.36·19-s + 1.52·21-s − 2.42·22-s − 4.52·23-s + 1.49·24-s − 5.62·27-s + 1.01·28-s + 6.38·29-s + 6.28·31-s + 32-s − 3.61·33-s − 5.04·34-s − 0.763·36-s + 1.80·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.863·3-s + 0.5·4-s + 0.610·6-s + 0.385·7-s + 0.353·8-s − 0.254·9-s − 0.729·11-s + 0.431·12-s + 0.272·14-s + 0.250·16-s − 1.22·17-s − 0.180·18-s − 1.68·19-s + 0.332·21-s − 0.515·22-s − 0.943·23-s + 0.305·24-s − 1.08·27-s + 0.192·28-s + 1.18·29-s + 1.12·31-s + 0.176·32-s − 0.630·33-s − 0.865·34-s − 0.127·36-s + 0.296·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 - 0.399T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 + 0.146T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 3.53T + 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 8.27T + 97T^{2} \) |
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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
−7.59596089942442378181031383297, −6.52489676138631889701044509360, −6.20493922587279050303681434531, −5.24970964222199208350951307679, −4.38884855102731753267313982580, −4.09964514234914736145383020601, −2.81801620015835015259326008395, −2.55563177816145732297825492160, −1.68371460417550622847748397221, 0,
1.68371460417550622847748397221, 2.55563177816145732297825492160, 2.81801620015835015259326008395, 4.09964514234914736145383020601, 4.38884855102731753267313982580, 5.24970964222199208350951307679, 6.20493922587279050303681434531, 6.52489676138631889701044509360, 7.59596089942442378181031383297
