L(s) = 1 | − 0.185·2-s − 3-s − 1.96·4-s + 1.26·5-s + 0.185·6-s + 0.737·8-s + 9-s − 0.234·10-s + 11-s + 1.96·12-s + 6.93·13-s − 1.26·15-s + 3.79·16-s − 5.60·17-s − 0.185·18-s − 5.75·19-s − 2.48·20-s − 0.185·22-s + 4.13·23-s − 0.737·24-s − 3.40·25-s − 1.28·26-s − 27-s − 6.09·29-s + 0.234·30-s + 5.30·31-s − 2.17·32-s + ⋯ |
L(s) = 1 | − 0.131·2-s − 0.577·3-s − 0.982·4-s + 0.564·5-s + 0.0758·6-s + 0.260·8-s + 0.333·9-s − 0.0742·10-s + 0.301·11-s + 0.567·12-s + 1.92·13-s − 0.326·15-s + 0.948·16-s − 1.36·17-s − 0.0438·18-s − 1.32·19-s − 0.555·20-s − 0.0396·22-s + 0.861·23-s − 0.150·24-s − 0.681·25-s − 0.252·26-s − 0.192·27-s − 1.13·29-s + 0.0428·30-s + 0.952·31-s − 0.385·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.136569585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136569585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 0.185T + 2T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 13 | \( 1 - 6.93T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 - 9.55T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−9.192802459770363803223370618838, −8.830617784380916206243921015209, −8.022533065910066859354693254723, −6.73688619343504748590370269701, −6.12520072590224000855260687645, −5.37972755374541224558996189727, −4.30095782427232958149108879345, −3.76962624284713139715448467457, −2.06340802795815706750767146859, −0.811005069325605831761243431284,
0.811005069325605831761243431284, 2.06340802795815706750767146859, 3.76962624284713139715448467457, 4.30095782427232958149108879345, 5.37972755374541224558996189727, 6.12520072590224000855260687645, 6.73688619343504748590370269701, 8.022533065910066859354693254723, 8.830617784380916206243921015209, 9.192802459770363803223370618838