L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 5·13-s − 15-s + 3·17-s − 2·19-s − 21-s − 6·23-s + 25-s + 5·27-s − 3·29-s − 4·31-s − 3·33-s + 35-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s − 2·45-s + 9·47-s + 49-s − 3·51-s − 12·53-s + 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.38·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.298·45-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.722556678528362158075028214833, −7.82866251649990434300481776759, −7.05288221116077551560339282808, −6.15246422175821288222036469083, −5.54934319897223270256186001671, −4.78822533627578751004226271654, −3.79583222083421885356258159776, −2.59983312100744179860207048134, −1.58268461056007998080671173620, 0,
1.58268461056007998080671173620, 2.59983312100744179860207048134, 3.79583222083421885356258159776, 4.78822533627578751004226271654, 5.54934319897223270256186001671, 6.15246422175821288222036469083, 7.05288221116077551560339282808, 7.82866251649990434300481776759, 8.722556678528362158075028214833