L(s) = 1 | + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 4·17-s + 2·23-s + 25-s + 27-s − 8·31-s − 33-s − 12·37-s + 2·39-s − 2·41-s + 12·43-s + 45-s − 4·51-s − 12·53-s − 55-s + 6·59-s + 10·61-s + 2·65-s + 8·67-s + 2·69-s − 8·71-s + 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.43·31-s − 0.174·33-s − 1.97·37-s + 0.320·39-s − 0.312·41-s + 1.82·43-s + 0.149·45-s − 0.560·51-s − 1.64·53-s − 0.134·55-s + 0.781·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s + 0.240·69-s − 0.949·71-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.83837355929473, −13.25101773048140, −12.85687436971885, −12.46481354158474, −11.86595440961708, −11.08645321473269, −10.86610622741266, −10.44394956110197, −9.728466811642369, −9.279644402533633, −8.894786105305710, −8.461533218401072, −7.879263212493456, −7.313974826501361, −6.789117169342659, −6.403603505432331, −5.644773869920494, −5.210652440833554, −4.670210115493731, −3.859449989551529, −3.572353865521882, −2.815959089729028, −2.177823715617141, −1.769002996287453, −0.9447183687712321, 0,
0.9447183687712321, 1.769002996287453, 2.177823715617141, 2.815959089729028, 3.572353865521882, 3.859449989551529, 4.670210115493731, 5.210652440833554, 5.644773869920494, 6.403603505432331, 6.789117169342659, 7.313974826501361, 7.879263212493456, 8.461533218401072, 8.894786105305710, 9.279644402533633, 9.728466811642369, 10.44394956110197, 10.86610622741266, 11.08645321473269, 11.86595440961708, 12.46481354158474, 12.85687436971885, 13.25101773048140, 13.83837355929473