L(s) = 1 | − 6·3-s + 7·7-s + 9·9-s + 56·11-s + 28·13-s + 90·17-s + 74·19-s − 42·21-s + 96·23-s + 108·27-s − 222·29-s − 100·31-s − 336·33-s − 58·37-s − 168·39-s + 422·41-s − 512·43-s − 148·47-s + 49·49-s − 540·51-s + 642·53-s − 444·57-s − 318·59-s + 720·61-s + 63·63-s + 412·67-s − 576·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.53·11-s + 0.597·13-s + 1.28·17-s + 0.893·19-s − 0.436·21-s + 0.870·23-s + 0.769·27-s − 1.42·29-s − 0.579·31-s − 1.77·33-s − 0.257·37-s − 0.689·39-s + 1.60·41-s − 1.81·43-s − 0.459·47-s + 1/7·49-s − 1.48·51-s + 1.66·53-s − 1.03·57-s − 0.701·59-s + 1.51·61-s + 0.125·63-s + 0.751·67-s − 1.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.800685677\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.800685677\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
good | 3 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 90 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 222 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 512 T + p^{3} T^{2} \) |
| 47 | \( 1 + 148 T + p^{3} T^{2} \) |
| 53 | \( 1 - 642 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 720 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 448 T + p^{3} T^{2} \) |
| 73 | \( 1 + 994 T + p^{3} T^{2} \) |
| 79 | \( 1 + 296 T + p^{3} T^{2} \) |
| 83 | \( 1 + 386 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 138 T + p^{3} T^{2} \) |
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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
−9.271012681888440505000952286931, −8.451530178566494117437963630480, −7.35804434446578196827846615106, −6.68855677817469967065022525726, −5.70804009137883443207701456346, −5.31803976701282148547032738683, −4.11831581581760763375177676807, −3.26914179255704336741879886573, −1.53376676615307360752000065980, −0.78551958475527996305957336665,
0.78551958475527996305957336665, 1.53376676615307360752000065980, 3.26914179255704336741879886573, 4.11831581581760763375177676807, 5.31803976701282148547032738683, 5.70804009137883443207701456346, 6.68855677817469967065022525726, 7.35804434446578196827846615106, 8.451530178566494117437963630480, 9.271012681888440505000952286931