| L(s) = 1 | + 0.0244·2-s − 3-s − 1.99·4-s + 5-s − 0.0244·6-s + 1.12·7-s − 0.0977·8-s + 9-s + 0.0244·10-s − 2.77·11-s + 1.99·12-s + 2.87·13-s + 0.0273·14-s − 15-s + 3.99·16-s + 7.29·17-s + 0.0244·18-s + 6.26·19-s − 1.99·20-s − 1.12·21-s − 0.0679·22-s + 0.0977·24-s + 25-s + 0.0702·26-s − 27-s − 2.24·28-s + 1.64·29-s + ⋯ |
| L(s) = 1 | + 0.0172·2-s − 0.577·3-s − 0.999·4-s + 0.447·5-s − 0.00998·6-s + 0.423·7-s − 0.0345·8-s + 0.333·9-s + 0.00773·10-s − 0.837·11-s + 0.577·12-s + 0.796·13-s + 0.00732·14-s − 0.258·15-s + 0.999·16-s + 1.76·17-s + 0.00576·18-s + 1.43·19-s − 0.447·20-s − 0.244·21-s − 0.0144·22-s + 0.0199·24-s + 0.200·25-s + 0.0137·26-s − 0.192·27-s − 0.423·28-s + 0.305·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801236804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801236804\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.0244T + 2T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 5.84T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.83959949412016363816757689110, −7.36499840716859848403846247741, −6.00796873093662288607408060528, −5.85198935822279363107540303022, −4.98257955302543880439247045502, −4.60438658434520564803343338853, −3.49641456530888082188847307290, −2.87112860198362954869538461640, −1.37131504083343425156349252882, −0.801474376327471627463806331905,
0.801474376327471627463806331905, 1.37131504083343425156349252882, 2.87112860198362954869538461640, 3.49641456530888082188847307290, 4.60438658434520564803343338853, 4.98257955302543880439247045502, 5.85198935822279363107540303022, 6.00796873093662288607408060528, 7.36499840716859848403846247741, 7.83959949412016363816757689110
