L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s − 2·13-s + 15-s + 16-s + 2·17-s − 18-s − 4·19-s − 20-s + 24-s + 25-s + 2·26-s − 27-s + 2·29-s − 30-s − 32-s − 2·34-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 − 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.23855566467195, −12.69750362313060, −12.47378455203179, −11.82653091368403, −11.58147578409187, −10.94259911919820, −10.62119774220307, −10.12503814777758, −9.652127471893764, −9.208861983219156, −8.445569722151538, −8.315880559722127, −7.537293463757933, −7.247861874764767, −6.685952304080626, −6.208037092757100, −5.639797452467933, −5.097040948762666, −4.542679114294792, −3.911313611703640, −3.444762932358302, −2.507868349624532, −2.237729522041523, −1.277807956869947, −0.7138653579295996, 0,
0.7138653579295996, 1.277807956869947, 2.237729522041523, 2.507868349624532, 3.444762932358302, 3.911313611703640, 4.542679114294792, 5.097040948762666, 5.639797452467933, 6.208037092757100, 6.685952304080626, 7.247861874764767, 7.537293463757933, 8.315880559722127, 8.445569722151538, 9.208861983219156, 9.652127471893764, 10.12503814777758, 10.62119774220307, 10.94259911919820, 11.58147578409187, 11.82653091368403, 12.47378455203179, 12.69750362313060, 13.23855566467195