L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s − 4·11-s − 12-s − 2·13-s − 14-s − 15-s − 16-s + 18-s + 20-s − 21-s − 4·22-s + 4·23-s − 3·24-s + 25-s − 2·26-s + 27-s + 28-s + 2·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.223·20-s − 0.218·21-s − 0.852·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9357178936\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9357178936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.01138949688032, −14.67777037328449, −13.99350632627525, −13.44916423815959, −13.13221537759840, −12.61937895891275, −12.14665183301526, −11.60778030831070, −10.66279101769890, −10.41405887681579, −9.604051022886420, −9.149545572629268, −8.623595085585358, −8.016303921975896, −7.460925828389479, −6.888553648579924, −6.158646429964338, −5.234176461008970, −5.117790232677420, −4.304773274649847, −3.681125383669039, −3.024901148841169, −2.671180646543710, −1.597281212917106, −0.3097468795299969,
0.3097468795299969, 1.597281212917106, 2.671180646543710, 3.024901148841169, 3.681125383669039, 4.304773274649847, 5.117790232677420, 5.234176461008970, 6.158646429964338, 6.888553648579924, 7.460925828389479, 8.016303921975896, 8.623595085585358, 9.149545572629268, 9.604051022886420, 10.41405887681579, 10.66279101769890, 11.60778030831070, 12.14665183301526, 12.61937895891275, 13.13221537759840, 13.44916423815959, 13.99350632627525, 14.67777037328449, 15.01138949688032