L(s) = 1 | − 3.37·5-s − 0.828i·11-s − 3.37i·13-s − 1.39·17-s + 6.75i·19-s − 2i·23-s + 6.41·25-s − 4.82i·29-s − 6.75i·31-s − 2.58·37-s − 8.15·41-s + 12.4·43-s + 6.75·47-s − 7.07i·53-s + 2.79i·55-s + ⋯ |
L(s) = 1 | − 1.51·5-s − 0.249i·11-s − 0.937i·13-s − 0.339·17-s + 1.55i·19-s − 0.417i·23-s + 1.28·25-s − 0.896i·29-s − 1.21i·31-s − 0.425·37-s − 1.27·41-s + 1.90·43-s + 0.985·47-s − 0.971i·53-s + 0.377i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1765134530\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1765134530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 - 6.75iT - 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 4.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.75iT - 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 8.15T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 6.75T + 59T^{2} \) |
| 61 | \( 1 - 8.15iT - 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 4.82iT - 71T^{2} \) |
| 73 | \( 1 - 1.39iT - 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 + 1.39iT - 97T^{2} \) |
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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
−8.133341375337232911795274007996, −7.65161473611829646923869581534, −6.99870509291527148129901174437, −6.01221287356759935072251979740, −5.48263038293373786777099821456, −4.36151740946110335605477083651, −3.94427879919725795316477955228, −3.22179582374106985687948057846, −2.26776952378465629430712287428, −0.882724409831705366553607951473,
0.05743720043484823956650769982, 1.29191832746272161862674408278, 2.51824631695324832158276186123, 3.35198298586698080503832491603, 4.13797751775257666169675402226, 4.65290829692977063682745093839, 5.39276507201394412616678059919, 6.57772722754232507757624799593, 7.12647919504995425818612047648, 7.46030339165219011867953745039