L(s) = 1 | + i·4-s − i·5-s + i·9-s + (−1 − i)11-s − 16-s + (−1 − i)19-s + 20-s − 25-s + i·29-s + (1 + i)31-s − 36-s + (1 − i)41-s + (1 − i)44-s + 45-s + 49-s + ⋯ |
L(s) = 1 | + i·4-s − i·5-s + i·9-s + (−1 − i)11-s − 16-s + (−1 − i)19-s + 20-s − 25-s + i·29-s + (1 + i)31-s − 36-s + (1 − i)41-s + (1 − i)44-s + 45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6128548394\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6128548394\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 29 | \( 1 - iT \) |
good | 2 | \( 1 - iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (1 + i)T + iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1 - i)T + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−13.19007158956853907013601278393, −12.60653029703028197500961535334, −11.40313183987310958416432208044, −10.47759928528073937459323626242, −8.766585948101750623257275695389, −8.365995397532859272406981295165, −7.21612790594965992491197968021, −5.46047728948515655924322566353, −4.37461038896300874501784182487, −2.66446834598452832199469494415,
2.37441829380017958106060039538, 4.23963785046997912056196187044, 5.85627077927153378918702552554, 6.66425983048885149547281387292, 7.929057424431366116466128070148, 9.608335148010529482036674461967, 10.15441551610510815076338042987, 11.10773304512960009381429154755, 12.24017337137894231430027189121, 13.42152706905986775804239387810