L(s) = 1 | − 1.97·3-s − 2.39·5-s − 7-s + 0.915·9-s + 2.34·11-s − 1.41·13-s + 4.74·15-s − 2.31·17-s − 5.93·19-s + 1.97·21-s − 6.90·23-s + 0.759·25-s + 4.12·27-s − 1.33·29-s + 4.98·31-s − 4.64·33-s + 2.39·35-s + 9.14·37-s + 2.79·39-s − 41-s + 9.18·43-s − 2.19·45-s + 10.6·47-s + 49-s + 4.58·51-s + 12.4·53-s − 5.63·55-s + ⋯ |
L(s) = 1 | − 1.14·3-s − 1.07·5-s − 0.377·7-s + 0.305·9-s + 0.708·11-s − 0.391·13-s + 1.22·15-s − 0.561·17-s − 1.36·19-s + 0.431·21-s − 1.43·23-s + 0.151·25-s + 0.793·27-s − 0.247·29-s + 0.894·31-s − 0.809·33-s + 0.405·35-s + 1.50·37-s + 0.447·39-s − 0.156·41-s + 1.40·43-s − 0.327·45-s + 1.55·47-s + 0.142·49-s + 0.641·51-s + 1.71·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5299328668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5299328668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 1.97T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 3.76T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−9.979918923796157901821669677985, −8.939882299689511127388472597409, −8.117583507626510376915501023454, −7.19063593704879584815657514247, −6.34023416478658622340701205837, −5.76159432702392286688156105835, −4.37864229851140824867293868672, −4.03997663152581777230283092434, −2.44948336092661583179636471914, −0.55744516357349007231923462071,
0.55744516357349007231923462071, 2.44948336092661583179636471914, 4.03997663152581777230283092434, 4.37864229851140824867293868672, 5.76159432702392286688156105835, 6.34023416478658622340701205837, 7.19063593704879584815657514247, 8.117583507626510376915501023454, 8.939882299689511127388472597409, 9.979918923796157901821669677985