L(s) = 1 | − 2·3-s + 3·5-s + 9-s + 2·13-s − 6·15-s + 17-s − 7·19-s − 3·23-s + 4·25-s + 4·27-s + 6·29-s + 4·31-s + 7·37-s − 4·39-s − 6·41-s + 43-s + 3·45-s − 2·51-s + 6·53-s + 14·57-s + 9·59-s + 14·61-s + 6·65-s − 5·67-s + 6·69-s − 15·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.34·5-s + 1/3·9-s + 0.554·13-s − 1.54·15-s + 0.242·17-s − 1.60·19-s − 0.625·23-s + 4/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.15·37-s − 0.640·39-s − 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.280·51-s + 0.824·53-s + 1.85·57-s + 1.17·59-s + 1.79·61-s + 0.744·65-s − 0.610·67-s + 0.722·69-s − 1.78·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.59104617191240, −14.18941988972573, −13.59561978843735, −13.12187282604513, −12.73040704651800, −12.11964343988004, −11.52943069615175, −11.22323468119281, −10.42980932955041, −10.08601363241038, −9.950281965600197, −8.824235264365414, −8.640522307119708, −8.001427426131667, −6.996531034005776, −6.551638056409799, −6.167176281216724, −5.642613902916327, −5.328583734921397, −4.398670940885561, −4.133395153423772, −2.911576731573686, −2.442923746372828, −1.610537967150409, −0.9711219220449954, 0,
0.9711219220449954, 1.610537967150409, 2.442923746372828, 2.911576731573686, 4.133395153423772, 4.398670940885561, 5.328583734921397, 5.642613902916327, 6.167176281216724, 6.551638056409799, 6.996531034005776, 8.001427426131667, 8.640522307119708, 8.824235264365414, 9.950281965600197, 10.08601363241038, 10.42980932955041, 11.22323468119281, 11.52943069615175, 12.11964343988004, 12.73040704651800, 13.12187282604513, 13.59561978843735, 14.18941988972573, 14.59104617191240