L(s) = 1 | + 3-s − 2·7-s + 9-s − 6·11-s + 2·13-s − 2·17-s − 4·19-s − 2·21-s + 27-s + 8·29-s − 6·33-s − 2·37-s + 2·39-s − 6·41-s − 43-s − 4·47-s − 3·49-s − 2·51-s + 2·53-s − 4·57-s − 6·59-s + 2·61-s − 2·63-s + 8·67-s + 12·71-s + 8·73-s + 12·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.436·21-s + 0.192·27-s + 1.48·29-s − 1.04·33-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.152·43-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.251·63-s + 0.977·67-s + 1.42·71-s + 0.936·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.14718574005315, −12.83148440704260, −12.66265837481968, −11.91482920786454, −11.31416346590388, −10.83538806113737, −10.32596175799954, −10.06511978384270, −9.616206102228196, −8.833560703697200, −8.515402139435599, −8.183584038878461, −7.618394025400694, −7.062211141212596, −6.453375202028535, −6.242816843952239, −5.443765274300856, −4.892709807096027, −4.542814593813164, −3.700722540589089, −3.316237662194611, −2.704724939774477, −2.306232287539763, −1.655096428249080, −0.6782793449189053, 0,
0.6782793449189053, 1.655096428249080, 2.306232287539763, 2.704724939774477, 3.316237662194611, 3.700722540589089, 4.542814593813164, 4.892709807096027, 5.443765274300856, 6.242816843952239, 6.453375202028535, 7.062211141212596, 7.618394025400694, 8.183584038878461, 8.515402139435599, 8.833560703697200, 9.616206102228196, 10.06511978384270, 10.32596175799954, 10.83538806113737, 11.31416346590388, 11.91482920786454, 12.66265837481968, 12.83148440704260, 13.14718574005315