| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 13-s − 15-s + 8·17-s − 7·19-s − 2·21-s + 25-s − 27-s − 7·29-s + 2·35-s + 10·37-s + 39-s + 3·41-s + 6·43-s + 45-s − 13·47-s − 3·49-s − 8·51-s + 53-s + 7·57-s − 2·59-s − 2·61-s + 2·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 1.94·17-s − 1.60·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.29·29-s + 0.338·35-s + 1.64·37-s + 0.160·39-s + 0.468·41-s + 0.914·43-s + 0.149·45-s − 1.89·47-s − 3/7·49-s − 1.12·51-s + 0.137·53-s + 0.927·57-s − 0.260·59-s − 0.256·61-s + 0.251·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.06409323584000, −12.86889193224942, −12.55454616033633, −11.86138640976456, −11.41343445612270, −11.06635454036622, −10.61302345106165, −9.919368853948949, −9.793116395906177, −9.191267453092253, −8.483498139098940, −8.067434599935869, −7.608353132644172, −7.163954064203227, −6.440495576542642, −5.986492715229965, −5.597218948609824, −5.115417860336879, −4.468105403651676, −4.129644448199875, −3.349415808217850, −2.728159977051418, −1.995522988196359, −1.523640084978481, −0.8732515704447253, 0,
0.8732515704447253, 1.523640084978481, 1.995522988196359, 2.728159977051418, 3.349415808217850, 4.129644448199875, 4.468105403651676, 5.115417860336879, 5.597218948609824, 5.986492715229965, 6.440495576542642, 7.163954064203227, 7.608353132644172, 8.067434599935869, 8.483498139098940, 9.191267453092253, 9.793116395906177, 9.919368853948949, 10.61302345106165, 11.06635454036622, 11.41343445612270, 11.86138640976456, 12.55454616033633, 12.86889193224942, 13.06409323584000
