| L(s) = 1 | + 2-s + 3-s + 4-s + 1.94·5-s + 6-s + 4.68·7-s + 8-s + 9-s + 1.94·10-s − 11-s + 12-s + 3.54·13-s + 4.68·14-s + 1.94·15-s + 16-s − 4.52·17-s + 18-s − 6.57·19-s + 1.94·20-s + 4.68·21-s − 22-s + 2.30·23-s + 24-s − 1.23·25-s + 3.54·26-s + 27-s + 4.68·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.867·5-s + 0.408·6-s + 1.77·7-s + 0.353·8-s + 0.333·9-s + 0.613·10-s − 0.301·11-s + 0.288·12-s + 0.982·13-s + 1.25·14-s + 0.501·15-s + 0.250·16-s − 1.09·17-s + 0.235·18-s − 1.50·19-s + 0.433·20-s + 1.02·21-s − 0.213·22-s + 0.481·23-s + 0.204·24-s − 0.246·25-s + 0.694·26-s + 0.192·27-s + 0.885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.540929907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.540929907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 1.94T + 5T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 2.30T + 23T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−8.306824034501664849880832046418, −7.956366603852677088733275844016, −6.74562368527471968132950920596, −6.26444124223212447142320245347, −5.31068973405183662170680249819, −4.59983185732770865449605827581, −4.10010974906145911256335538744, −2.79312609424135663538479251537, −2.04997348249997116548951699412, −1.39561615029384657038657488329,
1.39561615029384657038657488329, 2.04997348249997116548951699412, 2.79312609424135663538479251537, 4.10010974906145911256335538744, 4.59983185732770865449605827581, 5.31068973405183662170680249819, 6.26444124223212447142320245347, 6.74562368527471968132950920596, 7.956366603852677088733275844016, 8.306824034501664849880832046418
