L(s) = 1 | + 1.23·3-s + 2·5-s − 1.23·7-s − 1.47·9-s − 1.23·11-s + 4.47·13-s + 2.47·15-s + 17-s − 6.47·19-s − 1.52·21-s − 1.23·23-s − 25-s − 5.52·27-s + 2·29-s + 1.23·31-s − 1.52·33-s − 2.47·35-s − 10.9·37-s + 5.52·39-s + 2·41-s − 1.52·43-s − 2.94·45-s + 12.9·47-s − 5.47·49-s + 1.23·51-s − 2·53-s − 2.47·55-s + ⋯ |
L(s) = 1 | + 0.713·3-s + 0.894·5-s − 0.467·7-s − 0.490·9-s − 0.372·11-s + 1.24·13-s + 0.638·15-s + 0.242·17-s − 1.48·19-s − 0.333·21-s − 0.257·23-s − 0.200·25-s − 1.06·27-s + 0.371·29-s + 0.222·31-s − 0.265·33-s − 0.417·35-s − 1.79·37-s + 0.885·39-s + 0.312·41-s − 0.232·43-s − 0.438·45-s + 1.88·47-s − 0.781·49-s + 0.173·51-s − 0.274·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.374110438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.374110438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−13.49085759577834040588555307572, −12.44836080142256483393647294834, −10.99546039656749745576780395701, −10.05473493622270074119771615729, −8.942534572435227892849471776987, −8.213281007835140825652083695990, −6.55803287402778873278660169255, −5.59734670055123138599998759290, −3.69329493216429982385966860300, −2.24827141653266557516415079165,
2.24827141653266557516415079165, 3.69329493216429982385966860300, 5.59734670055123138599998759290, 6.55803287402778873278660169255, 8.213281007835140825652083695990, 8.942534572435227892849471776987, 10.05473493622270074119771615729, 10.99546039656749745576780395701, 12.44836080142256483393647294834, 13.49085759577834040588555307572