L(s) = 1 | + 0.414·2-s − 1.82·4-s + 5-s + 1.41·7-s − 1.58·8-s + 0.414·10-s − 6.24·11-s − 0.585·13-s + 0.585·14-s + 3·16-s − 6.82·17-s − 19-s − 1.82·20-s − 2.58·22-s + 3.65·23-s + 25-s − 0.242·26-s − 2.58·28-s + 1.41·29-s − 8.82·31-s + 4.41·32-s − 2.82·34-s + 1.41·35-s − 0.585·37-s − 0.414·38-s − 1.58·40-s − 8.24·41-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.447·5-s + 0.534·7-s − 0.560·8-s + 0.130·10-s − 1.88·11-s − 0.162·13-s + 0.156·14-s + 0.750·16-s − 1.65·17-s − 0.229·19-s − 0.408·20-s − 0.551·22-s + 0.762·23-s + 0.200·25-s − 0.0475·26-s − 0.488·28-s + 0.262·29-s − 1.58·31-s + 0.780·32-s − 0.485·34-s + 0.239·35-s − 0.0963·37-s − 0.0671·38-s − 0.250·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 0.585T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−9.722745648379049918646211507621, −8.866871035962801030941946926719, −8.214413845173848281060607772219, −7.25258869154653640498590323148, −6.04254267696844185646324438513, −5.03561476211193453184203592506, −4.69800476778969242459409458966, −3.24134624149838719801586204400, −2.06707568249052481512345127651, 0,
2.06707568249052481512345127651, 3.24134624149838719801586204400, 4.69800476778969242459409458966, 5.03561476211193453184203592506, 6.04254267696844185646324438513, 7.25258869154653640498590323148, 8.214413845173848281060607772219, 8.866871035962801030941946926719, 9.722745648379049918646211507621