L(s) = 1 | − 2-s + 0.455·3-s + 4-s − 2.88·5-s − 0.455·6-s − 3.99·7-s − 8-s − 2.79·9-s + 2.88·10-s + 2.79·11-s + 0.455·12-s + 4.19·13-s + 3.99·14-s − 1.31·15-s + 16-s + 7.24·17-s + 2.79·18-s − 3.58·19-s − 2.88·20-s − 1.81·21-s − 2.79·22-s − 23-s − 0.455·24-s + 3.29·25-s − 4.19·26-s − 2.63·27-s − 3.99·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.262·3-s + 0.5·4-s − 1.28·5-s − 0.185·6-s − 1.50·7-s − 0.353·8-s − 0.930·9-s + 0.910·10-s + 0.841·11-s + 0.131·12-s + 1.16·13-s + 1.06·14-s − 0.338·15-s + 0.250·16-s + 1.75·17-s + 0.658·18-s − 0.823·19-s − 0.644·20-s − 0.396·21-s − 0.595·22-s − 0.208·23-s − 0.0928·24-s + 0.659·25-s − 0.822·26-s − 0.507·27-s − 0.754·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4737024392\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4737024392\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 0.455T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 7.24T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 0.209T + 37T^{2} \) |
| 41 | \( 1 + 3.08T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 7.77T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−8.112766710459223603687791803904, −7.53426102123057079414352219795, −6.79315837786483954032297109630, −6.05915746145707196190431302699, −5.53233067589221437223096405454, −3.85173481511326558702983735540, −3.63610581682495696177338052928, −3.04872355073050158472926701243, −1.67933122846689493090921686816, −0.39082896641018334595395177483,
0.39082896641018334595395177483, 1.67933122846689493090921686816, 3.04872355073050158472926701243, 3.63610581682495696177338052928, 3.85173481511326558702983735540, 5.53233067589221437223096405454, 6.05915746145707196190431302699, 6.79315837786483954032297109630, 7.53426102123057079414352219795, 8.112766710459223603687791803904