L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 11·5-s − 6·6-s − 15·7-s + 8·8-s + 9·9-s − 22·10-s − 29·11-s − 12·12-s − 82·13-s − 30·14-s + 33·15-s + 16·16-s + 27·17-s + 18·18-s − 19·19-s − 44·20-s + 45·21-s − 58·22-s + 100·23-s − 24·24-s − 4·25-s − 164·26-s − 27·27-s − 60·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.983·5-s − 0.408·6-s − 0.809·7-s + 0.353·8-s + 1/3·9-s − 0.695·10-s − 0.794·11-s − 0.288·12-s − 1.74·13-s − 0.572·14-s + 0.568·15-s + 1/4·16-s + 0.385·17-s + 0.235·18-s − 0.229·19-s − 0.491·20-s + 0.467·21-s − 0.562·22-s + 0.906·23-s − 0.204·24-s − 0.0319·25-s − 1.23·26-s − 0.192·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 + 29 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 23 | \( 1 - 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 118 T + p^{3} T^{2} \) |
| 31 | \( 1 - 70 T + p^{3} T^{2} \) |
| 37 | \( 1 - 232 T + p^{3} T^{2} \) |
| 41 | \( 1 - 8 T + p^{3} T^{2} \) |
| 43 | \( 1 + 287 T + p^{3} T^{2} \) |
| 47 | \( 1 - 385 T + p^{3} T^{2} \) |
| 53 | \( 1 - 538 T + p^{3} T^{2} \) |
| 59 | \( 1 + 300 T + p^{3} T^{2} \) |
| 61 | \( 1 + 901 T + p^{3} T^{2} \) |
| 67 | \( 1 - 132 T + p^{3} T^{2} \) |
| 71 | \( 1 - 472 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1131 T + p^{3} T^{2} \) |
| 79 | \( 1 + 52 T + p^{3} T^{2} \) |
| 83 | \( 1 - 276 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1310 T + p^{3} T^{2} \) |
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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
−12.48558259068143784229306433961, −11.83923050311612832089209773902, −10.70221095137149637260144412373, −9.636197444201825391359013721852, −7.78437732274967728659003527739, −6.96311057620402844247045906430, −5.51328713805008924068239116456, −4.37285940591065295095702319728, −2.87291286165890869529780142721, 0,
2.87291286165890869529780142721, 4.37285940591065295095702319728, 5.51328713805008924068239116456, 6.96311057620402844247045906430, 7.78437732274967728659003527739, 9.636197444201825391359013721852, 10.70221095137149637260144412373, 11.83923050311612832089209773902, 12.48558259068143784229306433961