| L(s) = 1 | − 2.19·3-s − 1.70·5-s + 7-s + 1.81·9-s − 4.56·11-s − 13-s + 3.74·15-s + 7.37·17-s − 6.46·19-s − 2.19·21-s − 5.90·23-s − 2.08·25-s + 2.59·27-s + 2.69·29-s − 0.523·31-s + 10.0·33-s − 1.70·35-s + 2.17·37-s + 2.19·39-s − 4.98·41-s − 2.32·43-s − 3.10·45-s + 12.3·47-s + 49-s − 16.1·51-s − 4.71·53-s + 7.78·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.763·5-s + 0.377·7-s + 0.605·9-s − 1.37·11-s − 0.277·13-s + 0.967·15-s + 1.78·17-s − 1.48·19-s − 0.478·21-s − 1.23·23-s − 0.417·25-s + 0.500·27-s + 0.500·29-s − 0.0939·31-s + 1.74·33-s − 0.288·35-s + 0.357·37-s + 0.351·39-s − 0.778·41-s − 0.355·43-s − 0.462·45-s + 1.79·47-s + 0.142·49-s − 2.26·51-s − 0.648·53-s + 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5826238275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5826238275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 + 0.523T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 0.137T + 61T^{2} \) |
| 67 | \( 1 - 7.96T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 7.51T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−9.905825045000487672407454853295, −8.348459132699610656178341678768, −7.981093066797541115918917676045, −7.10417507988091395850649500491, −6.02817598516318060766268215897, −5.42465997235892209931953524534, −4.64882441792769940626794384977, −3.65529121232184421737818126557, −2.27542611612669956223882072746, −0.55919019417946872945679965196,
0.55919019417946872945679965196, 2.27542611612669956223882072746, 3.65529121232184421737818126557, 4.64882441792769940626794384977, 5.42465997235892209931953524534, 6.02817598516318060766268215897, 7.10417507988091395850649500491, 7.981093066797541115918917676045, 8.348459132699610656178341678768, 9.905825045000487672407454853295
