L(s) = 1 | − 2-s + 3-s − 4-s + 4·5-s − 6-s − 7-s + 3·8-s + 9-s − 4·10-s − 2·11-s − 12-s + 4·13-s + 14-s + 4·15-s − 16-s − 18-s − 19-s − 4·20-s − 21-s + 2·22-s − 6·23-s + 3·24-s + 11·25-s − 4·26-s + 27-s + 28-s + 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 1.26·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.894·20-s − 0.218·21-s + 0.426·22-s − 1.25·23-s + 0.612·24-s + 11/5·25-s − 0.784·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.333118326\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333118326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.59977053409892285586037254314, −10.24450631461714262215841715867, −9.396900065180270074056113073813, −8.763637556939305839714412297013, −7.907151107890200846241427284533, −6.50454409559776666396301031990, −5.65610600854513533130990909511, −4.35778394220277191815347788708, −2.73042548756576274662693553032, −1.42946173088223678997984816291,
1.42946173088223678997984816291, 2.73042548756576274662693553032, 4.35778394220277191815347788708, 5.65610600854513533130990909511, 6.50454409559776666396301031990, 7.907151107890200846241427284533, 8.763637556939305839714412297013, 9.396900065180270074056113073813, 10.24450631461714262215841715867, 10.59977053409892285586037254314