L(s) = 1 | − 0.0357·2-s + 3·3-s − 7.99·4-s − 7.80·5-s − 0.107·6-s + 4.90·7-s + 0.571·8-s + 9·9-s + 0.279·10-s + 37.9·11-s − 23.9·12-s + 27.2·13-s − 0.175·14-s − 23.4·15-s + 63.9·16-s + 63.3·17-s − 0.321·18-s + 97.9·19-s + 62.4·20-s + 14.7·21-s − 1.35·22-s + 9.58·23-s + 1.71·24-s − 64.0·25-s − 0.972·26-s + 27·27-s − 39.2·28-s + ⋯ |
L(s) = 1 | − 0.0126·2-s + 0.577·3-s − 0.999·4-s − 0.698·5-s − 0.00729·6-s + 0.264·7-s + 0.0252·8-s + 0.333·9-s + 0.00882·10-s + 1.04·11-s − 0.577·12-s + 0.580·13-s − 0.00334·14-s − 0.403·15-s + 0.999·16-s + 0.903·17-s − 0.00421·18-s + 1.18·19-s + 0.698·20-s + 0.152·21-s − 0.0131·22-s + 0.0868·23-s + 0.0145·24-s − 0.512·25-s − 0.00733·26-s + 0.192·27-s − 0.264·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.649557639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.649557639\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
good | 2 | \( 1 + 0.0357T + 8T^{2} \) |
| 5 | \( 1 + 7.80T + 125T^{2} \) |
| 7 | \( 1 - 4.90T + 343T^{2} \) |
| 11 | \( 1 - 37.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.58T + 1.21e4T^{2} \) |
| 29 | \( 1 - 65.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 77.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 558.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 529.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 393.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 50.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−12.25869342976133780538501383282, −11.39280411579479645202949987316, −9.960618572997652833976388168860, −9.130507397620017230457133962579, −8.228959727562557878744335020676, −7.34856797747743420430299454913, −5.65762029449133554802796713064, −4.24570465544780806227149436518, −3.42890203885916183776586359926, −1.08595591761035775567808517584,
1.08595591761035775567808517584, 3.42890203885916183776586359926, 4.24570465544780806227149436518, 5.65762029449133554802796713064, 7.34856797747743420430299454913, 8.228959727562557878744335020676, 9.130507397620017230457133962579, 9.960618572997652833976388168860, 11.39280411579479645202949987316, 12.25869342976133780538501383282