L(s) = 1 | + 2·5-s + 4·7-s − 4·11-s − 13-s − 2·17-s − 25-s − 10·29-s − 4·31-s + 8·35-s + 2·37-s − 6·41-s − 12·43-s + 9·49-s + 6·53-s − 8·55-s − 12·59-s + 2·61-s − 2·65-s − 8·67-s + 2·73-s − 16·77-s − 8·79-s − 4·83-s − 4·85-s + 2·89-s − 4·91-s + 10·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 1/5·25-s − 1.85·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.937·41-s − 1.82·43-s + 9/7·49-s + 0.824·53-s − 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.234·73-s − 1.82·77-s − 0.900·79-s − 0.439·83-s − 0.433·85-s + 0.211·89-s − 0.419·91-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.62853663834564851839544876819, −6.97253769063291899213365954407, −5.93855747234204168097494026985, −5.35091971311059042197097159320, −4.93913154418367326541339053900, −4.06970235811330699263085257364, −2.96431053600696331099457094648, −1.97956305248263035021860674125, −1.66905766688203639375309352595, 0,
1.66905766688203639375309352595, 1.97956305248263035021860674125, 2.96431053600696331099457094648, 4.06970235811330699263085257364, 4.93913154418367326541339053900, 5.35091971311059042197097159320, 5.93855747234204168097494026985, 6.97253769063291899213365954407, 7.62853663834564851839544876819