L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 6·13-s − 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 24-s + 25-s + 6·26-s − 27-s − 6·29-s + 30-s − 32-s + 34-s + 36-s − 2·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.71642569047836, −14.18978949496571, −13.30497816920548, −12.98743054158162, −12.38831985018434, −12.04119589054800, −11.31388985782596, −11.01256296941139, −10.37230328016918, −9.899357727043876, −9.541655463610307, −9.006631701858411, −8.338960223641259, −7.787565161989552, −7.178541111308336, −6.773237801530766, −6.265681141614419, −5.399851804291723, −5.267815461915840, −4.367155495530454, −3.836156108145628, −2.802200143204851, −2.258366507571343, −1.753834629739340, −0.7282894718276465, 0,
0.7282894718276465, 1.753834629739340, 2.258366507571343, 2.802200143204851, 3.836156108145628, 4.367155495530454, 5.267815461915840, 5.399851804291723, 6.265681141614419, 6.773237801530766, 7.178541111308336, 7.787565161989552, 8.338960223641259, 9.006631701858411, 9.541655463610307, 9.899357727043876, 10.37230328016918, 11.01256296941139, 11.31388985782596, 12.04119589054800, 12.38831985018434, 12.98743054158162, 13.30497816920548, 14.18978949496571, 14.71642569047836