| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 12-s + 13-s + 2·15-s + 16-s − 18-s + 2·20-s − 24-s − 25-s − 26-s + 27-s + 2·29-s − 2·30-s − 8·31-s − 32-s + 36-s − 10·37-s + 39-s − 2·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.447·20-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s − 1.43·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 0.160·39-s − 0.316·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22542 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22542 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.82599960582097, −15.28732116203472, −14.65904030381757, −14.14682950671206, −13.74728690634155, −13.02051175369603, −12.65890339011116, −11.94938085097111, −11.18465241460535, −10.79477013030339, −10.05864656071574, −9.666986454021233, −9.203176797362991, −8.581099684546272, −8.083813899130498, −7.462276174169152, −6.700226674982731, −6.362945326893949, −5.448370226686127, −5.039064927468432, −3.898827539159015, −3.377371437222914, −2.492312477121279, −1.878802386365631, −1.279271973995709, 0,
1.279271973995709, 1.878802386365631, 2.492312477121279, 3.377371437222914, 3.898827539159015, 5.039064927468432, 5.448370226686127, 6.362945326893949, 6.700226674982731, 7.462276174169152, 8.083813899130498, 8.581099684546272, 9.203176797362991, 9.666986454021233, 10.05864656071574, 10.79477013030339, 11.18465241460535, 11.94938085097111, 12.65890339011116, 13.02051175369603, 13.74728690634155, 14.14682950671206, 14.65904030381757, 15.28732116203472, 15.82599960582097
