| L(s) = 1 | + 2-s − 3-s + 4-s − 3.99·5-s − 6-s − 2.56·7-s + 8-s + 9-s − 3.99·10-s − 2.60·11-s − 12-s − 13-s − 2.56·14-s + 3.99·15-s + 16-s + 2.12·17-s + 18-s + 0.868·19-s − 3.99·20-s + 2.56·21-s − 2.60·22-s + 5.16·23-s − 24-s + 10.9·25-s − 26-s − 27-s − 2.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.78·5-s − 0.408·6-s − 0.971·7-s + 0.353·8-s + 0.333·9-s − 1.26·10-s − 0.784·11-s − 0.288·12-s − 0.277·13-s − 0.686·14-s + 1.03·15-s + 0.250·16-s + 0.516·17-s + 0.235·18-s + 0.199·19-s − 0.892·20-s + 0.560·21-s − 0.554·22-s + 1.07·23-s − 0.204·24-s + 2.18·25-s − 0.196·26-s − 0.192·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 3.99T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 - 0.868T + 19T^{2} \) |
| 23 | \( 1 - 5.16T + 23T^{2} \) |
| 29 | \( 1 + 0.693T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 0.0582T + 47T^{2} \) |
| 53 | \( 1 + 0.983T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 - 2.12T + 61T^{2} \) |
| 67 | \( 1 - 3.67T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.44890370882826649598817378530, −6.77294688987806467856272813105, −6.14576011719782336346856214630, −5.18079236715036852873471248316, −4.70844326550418590165199831856, −3.90762849596744759594013166079, −3.24531335593843130294533623436, −2.67336309263929383748080997621, −0.979063179060467373095483688087, 0,
0.979063179060467373095483688087, 2.67336309263929383748080997621, 3.24531335593843130294533623436, 3.90762849596744759594013166079, 4.70844326550418590165199831856, 5.18079236715036852873471248316, 6.14576011719782336346856214630, 6.77294688987806467856272813105, 7.44890370882826649598817378530
