L(s) = 1 | − 1.05·3-s + 1.77·5-s + 3.57·7-s − 1.89·9-s + 1.62·11-s + 6.40·13-s − 1.86·15-s − 0.871·17-s + 6.98·19-s − 3.76·21-s + 1.58·23-s − 1.86·25-s + 5.15·27-s − 1.81·29-s − 2.87·31-s − 1.71·33-s + 6.32·35-s − 3.69·37-s − 6.75·39-s + 2.10·41-s − 43-s − 3.34·45-s + 1.24·47-s + 5.76·49-s + 0.918·51-s − 0.645·53-s + 2.88·55-s + ⋯ |
L(s) = 1 | − 0.608·3-s + 0.792·5-s + 1.35·7-s − 0.630·9-s + 0.490·11-s + 1.77·13-s − 0.481·15-s − 0.211·17-s + 1.60·19-s − 0.821·21-s + 0.331·23-s − 0.372·25-s + 0.991·27-s − 0.336·29-s − 0.515·31-s − 0.298·33-s + 1.06·35-s − 0.608·37-s − 1.08·39-s + 0.328·41-s − 0.152·43-s − 0.499·45-s + 0.181·47-s + 0.823·49-s + 0.128·51-s − 0.0886·53-s + 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.257036733\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.257036733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 0.871T + 17T^{2} \) |
| 19 | \( 1 - 6.98T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 + 0.645T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 7.95T + 83T^{2} \) |
| 89 | \( 1 - 0.475T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.796986820330875817783761164561, −8.187238840768769344011008146281, −7.26691748588971131210307387575, −6.31945601814925505487604079746, −5.62380779443264477641079453046, −5.24024209059405320037184205486, −4.14281191971226111277224704946, −3.14680277104711927118256795841, −1.80531816513911318359509658413, −1.07617644546271927239708482235,
1.07617644546271927239708482235, 1.80531816513911318359509658413, 3.14680277104711927118256795841, 4.14281191971226111277224704946, 5.24024209059405320037184205486, 5.62380779443264477641079453046, 6.31945601814925505487604079746, 7.26691748588971131210307387575, 8.187238840768769344011008146281, 8.796986820330875817783761164561