L(s) = 1 | + 3-s + 9-s + 4·13-s + 4·17-s − 4·19-s + 4·23-s − 5·25-s + 27-s + 2·29-s + 8·31-s − 6·37-s + 4·39-s + 12·41-s + 4·43-s − 8·47-s + 4·51-s + 6·53-s − 4·57-s + 12·59-s + 4·61-s − 4·67-s + 4·69-s − 12·71-s + 8·73-s − 5·75-s − 16·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.10·13-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.640·39-s + 1.87·41-s + 0.609·43-s − 1.16·47-s + 0.560·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.512·61-s − 0.488·67-s + 0.481·69-s − 1.42·71-s + 0.936·73-s − 0.577·75-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.161595621\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.161595621\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.783168027646447439541954623254, −8.797985223964531881327538956416, −8.282758185006394553526891603643, −7.41005169816589214200697437145, −6.43859371104521209533782451242, −5.62427172231298026532892197535, −4.40915109577399770914652954180, −3.57155516860156643593206388626, −2.52790453796007379380617788419, −1.17836444974596423542040449778,
1.17836444974596423542040449778, 2.52790453796007379380617788419, 3.57155516860156643593206388626, 4.40915109577399770914652954180, 5.62427172231298026532892197535, 6.43859371104521209533782451242, 7.41005169816589214200697437145, 8.282758185006394553526891603643, 8.797985223964531881327538956416, 9.783168027646447439541954623254