| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 13-s + 15-s + 16-s − 6·17-s + 18-s − 20-s − 4·23-s − 24-s + 25-s + 26-s − 27-s − 2·29-s + 30-s + 32-s − 6·34-s + 36-s + 6·37-s − 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834858567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834858567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.61924474338355, −14.00150514247831, −13.54581678026820, −12.93395591135847, −12.62814814699481, −12.04140738881432, −11.34978723301856, −11.24936274609849, −10.65220493714023, −10.06162579995303, −9.348077013262253, −8.869154057525022, −8.031977632909496, −7.699729143560607, −6.926186578775066, −6.497215330745520, −5.942549695198086, −5.438790116365138, −4.551738336366413, −4.351088221473278, −3.714505779632711, −2.900417731283327, −2.216206288292623, −1.459681993217824, −0.4387714611955148,
0.4387714611955148, 1.459681993217824, 2.216206288292623, 2.900417731283327, 3.714505779632711, 4.351088221473278, 4.551738336366413, 5.438790116365138, 5.942549695198086, 6.497215330745520, 6.926186578775066, 7.699729143560607, 8.031977632909496, 8.869154057525022, 9.348077013262253, 10.06162579995303, 10.65220493714023, 11.24936274609849, 11.34978723301856, 12.04140738881432, 12.62814814699481, 12.93395591135847, 13.54581678026820, 14.00150514247831, 14.61924474338355
