| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 5-s − 1.41·6-s − 0.414·7-s − 8-s − 0.999·9-s + 10-s − 3·11-s + 1.41·12-s − 4.24·13-s + 0.414·14-s − 1.41·15-s + 16-s − 0.585·17-s + 0.999·18-s − 4.82·19-s − 20-s − 0.585·21-s + 3·22-s − 1.41·24-s + 25-s + 4.24·26-s − 5.65·27-s − 0.414·28-s − 3.41·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.447·5-s − 0.577·6-s − 0.156·7-s − 0.353·8-s − 0.333·9-s + 0.316·10-s − 0.904·11-s + 0.408·12-s − 1.17·13-s + 0.110·14-s − 0.365·15-s + 0.250·16-s − 0.142·17-s + 0.235·18-s − 1.10·19-s − 0.223·20-s − 0.127·21-s + 0.639·22-s − 0.288·24-s + 0.200·25-s + 0.832·26-s − 1.08·27-s − 0.0782·28-s − 0.634·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
| 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
−9.890308502181652248913660196336, −9.185093532110060905282803485733, −8.234516391428505055729235144580, −7.79769465064566373879848289204, −6.83808283418511137949785375695, −5.62377957018929699234442023327, −4.35558102699223998466485077843, −3.00163802097250155834066457543, −2.23247508298351041078120036435, 0,
2.23247508298351041078120036435, 3.00163802097250155834066457543, 4.35558102699223998466485077843, 5.62377957018929699234442023327, 6.83808283418511137949785375695, 7.79769465064566373879848289204, 8.234516391428505055729235144580, 9.185093532110060905282803485733, 9.890308502181652248913660196336
