| L(s) = 1 | + 3-s − i·5-s − i·7-s + i·11-s − i·15-s − i·21-s − 25-s − 27-s + (1 + i)31-s + i·33-s − 35-s − i·37-s + i·41-s + (1 + i)43-s + i·47-s + ⋯ |
| L(s) = 1 | + 3-s − i·5-s − i·7-s + i·11-s − i·15-s − i·21-s − 25-s − 27-s + (1 + i)31-s + i·33-s − 35-s − i·37-s + i·41-s + (1 + i)43-s + i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246664927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246664927\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-1 + i)T - iT^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 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
−10.25568745673041232392408024981, −9.480101897896462060185475216704, −8.831829086715782214702007489982, −7.888967216271389877968668404181, −7.37184725362495879523472201970, −6.07874611895756882370790525401, −4.74968310816742776401947003273, −4.11579110887827576871678713366, −2.85769584937893753582032870896, −1.46806011096337313505749075438,
2.28419346217152213566924936899, 2.94998505337625106834770821887, 3.87989686433611595479549379133, 5.53143500709999724914375613130, 6.22516514386988278338421210863, 7.34791937619071667550532329305, 8.287091089627404246905712027391, 8.815956066759734611264887578989, 9.709738251650673909448266769049, 10.63325544955693430019050879680
