| L(s) = 1 | + 2·3-s − 5-s + 9-s − 2·13-s − 2·15-s + 6·17-s + 2·23-s + 25-s − 4·27-s − 8·29-s − 4·31-s − 2·37-s − 4·39-s − 4·41-s + 4·43-s − 45-s − 2·47-s − 7·49-s + 12·51-s − 10·53-s + 8·61-s + 2·65-s + 2·67-s + 4·69-s − 8·71-s − 10·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.417·23-s + 1/5·25-s − 0.769·27-s − 1.48·29-s − 0.718·31-s − 0.328·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s − 0.149·45-s − 0.291·47-s − 49-s + 1.68·51-s − 1.37·53-s + 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.481·69-s − 0.949·71-s − 1.17·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54811114930240227810952805544, −6.96026653359103287180014228525, −5.88479031484423197355353637309, −5.28193506050731343782060552673, −4.42825460198157208676674822743, −3.45238664618383510275842919404, −3.27022364965941645073353440498, −2.26588884063590301804734609180, −1.42479578511137833146794193415, 0,
1.42479578511137833146794193415, 2.26588884063590301804734609180, 3.27022364965941645073353440498, 3.45238664618383510275842919404, 4.42825460198157208676674822743, 5.28193506050731343782060552673, 5.88479031484423197355353637309, 6.96026653359103287180014228525, 7.54811114930240227810952805544
