L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 3·13-s + 16-s − 17-s − 7·19-s + 6·22-s + 8·23-s + 3·26-s + 5·29-s + 5·31-s + 32-s − 34-s + 8·37-s − 7·38-s − 4·43-s + 6·44-s + 8·46-s − 3·47-s − 7·49-s + 3·52-s − 9·53-s + 5·58-s − 5·59-s − 3·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 0.832·13-s + 1/4·16-s − 0.242·17-s − 1.60·19-s + 1.27·22-s + 1.66·23-s + 0.588·26-s + 0.928·29-s + 0.898·31-s + 0.176·32-s − 0.171·34-s + 1.31·37-s − 1.13·38-s − 0.609·43-s + 0.904·44-s + 1.17·46-s − 0.437·47-s − 49-s + 0.416·52-s − 1.23·53-s + 0.656·58-s − 0.650·59-s − 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.103661850\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.103661850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−7.86625989498839697894186665041, −6.71978821832837603727796473612, −6.52921660631650771229096311130, −6.00423531650877286662157691066, −4.75275142198392183299279007503, −4.44319372834745597801581895719, −3.59130951406283393164894958848, −2.90022234396185536904625056998, −1.78274366372577389926427627202, −0.979933555115995156564765897609,
0.979933555115995156564765897609, 1.78274366372577389926427627202, 2.90022234396185536904625056998, 3.59130951406283393164894958848, 4.44319372834745597801581895719, 4.75275142198392183299279007503, 6.00423531650877286662157691066, 6.52921660631650771229096311130, 6.71978821832837603727796473612, 7.86625989498839697894186665041