L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3·11-s + 3·13-s + 15-s + 17-s − 3·19-s + 21-s − 7·23-s − 4·25-s + 27-s − 6·29-s − 10·31-s − 3·33-s + 35-s + 4·37-s + 3·39-s − 9·41-s − 9·43-s + 45-s − 6·47-s + 49-s + 51-s − 10·53-s − 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.832·13-s + 0.258·15-s + 0.242·17-s − 0.688·19-s + 0.218·21-s − 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.522·33-s + 0.169·35-s + 0.657·37-s + 0.480·39-s − 1.40·41-s − 1.37·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.140·51-s − 1.37·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.047510397795042759306435745334, −7.16816928068587149939416718933, −6.27589350201497287047172248962, −5.62987184097806277263758647807, −4.92022376297891425412602607399, −3.88150733045739589604318636817, −3.35928695624332686054224401673, −2.07938171680371245897138454958, −1.75286629305681237514154200662, 0,
1.75286629305681237514154200662, 2.07938171680371245897138454958, 3.35928695624332686054224401673, 3.88150733045739589604318636817, 4.92022376297891425412602607399, 5.62987184097806277263758647807, 6.27589350201497287047172248962, 7.16816928068587149939416718933, 8.047510397795042759306435745334