L(s) = 1 | − i·3-s − i·4-s − 9-s − i·11-s − 12-s − 16-s + (1 + i)23-s + i·27-s − 33-s + i·36-s + (−1 − i)37-s − 44-s + (1 − i)47-s + i·48-s + i·49-s + ⋯ |
L(s) = 1 | − i·3-s − i·4-s − 9-s − i·11-s − 12-s − 16-s + (1 + i)23-s + i·27-s − 33-s + i·36-s + (−1 − i)37-s − 44-s + (1 − i)47-s + i·48-s + i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9024301994\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9024301994\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−10.23071667737376583706159872533, −9.147084726437311251292084534402, −8.549687894464285005380232598372, −7.41566701458488614582239574045, −6.66968579339465231041763409932, −5.75982551728490804295116825813, −5.18191648475780262883773136600, −3.52770520345652397958764540371, −2.22687602096433717607873770995, −0.968622543135008780305757496039,
2.43388220621920797108667822332, 3.43420099191489485531905242408, 4.41192657316577392882689420804, 5.07739709847711050946444095747, 6.47753792155094213698673409059, 7.35848124263822718740617964661, 8.362712811836855626608715311180, 8.997557921523313180826912865509, 9.851895680173775052036593960974, 10.66610716362622758982462772614