L(s) = 1 | + 5-s + 3·7-s − 3·9-s + 6·11-s − 4·13-s − 5·19-s − 4·23-s − 4·25-s + 2·29-s − 31-s + 3·35-s − 2·37-s − 9·41-s + 2·43-s − 3·45-s + 4·47-s + 2·49-s + 12·53-s + 6·55-s + 9·59-s + 12·61-s − 9·63-s − 4·65-s − 12·67-s + 5·71-s − 14·73-s + 18·77-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 9-s + 1.80·11-s − 1.10·13-s − 1.14·19-s − 0.834·23-s − 4/5·25-s + 0.371·29-s − 0.179·31-s + 0.507·35-s − 0.328·37-s − 1.40·41-s + 0.304·43-s − 0.447·45-s + 0.583·47-s + 2/7·49-s + 1.64·53-s + 0.809·55-s + 1.17·59-s + 1.53·61-s − 1.13·63-s − 0.496·65-s − 1.46·67-s + 0.593·71-s − 1.63·73-s + 2.05·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.175529561\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175529561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.67571434204969194022003943130, −12.03962739422248223254555347561, −11.61492332447501858533901012493, −10.30281769176055663000387399267, −9.092450053143345820975212938188, −8.206280855235405828971635271957, −6.74587165045419703295250646596, −5.52489137298948854722822938762, −4.14055388391449335150730893666, −2.04911425078546165359323933060,
2.04911425078546165359323933060, 4.14055388391449335150730893666, 5.52489137298948854722822938762, 6.74587165045419703295250646596, 8.206280855235405828971635271957, 9.092450053143345820975212938188, 10.30281769176055663000387399267, 11.61492332447501858533901012493, 12.03962739422248223254555347561, 13.67571434204969194022003943130