| L(s) = 1 | − 2-s − 2.73·3-s + 4-s + 1.73·5-s + 2.73·6-s − 1.26·7-s − 8-s + 4.46·9-s − 1.73·10-s − 2.73·12-s − 3·13-s + 1.26·14-s − 4.73·15-s + 16-s − 5.19·17-s − 4.46·18-s − 4.73·19-s + 1.73·20-s + 3.46·21-s − 4.73·23-s + 2.73·24-s − 2.00·25-s + 3·26-s − 3.99·27-s − 1.26·28-s + 3·29-s + 4.73·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.774·5-s + 1.11·6-s − 0.479·7-s − 0.353·8-s + 1.48·9-s − 0.547·10-s − 0.788·12-s − 0.832·13-s + 0.338·14-s − 1.22·15-s + 0.250·16-s − 1.26·17-s − 1.05·18-s − 1.08·19-s + 0.387·20-s + 0.755·21-s − 0.986·23-s + 0.557·24-s − 0.400·25-s + 0.588·26-s − 0.769·27-s − 0.239·28-s + 0.557·29-s + 0.863·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 0.803T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 0.464T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−11.41655744947857510307868720604, −10.65492981650156073391598535896, −9.907647018896641532019228200236, −9.000215228200752369436797457893, −7.42021390799642292132016013162, −6.37513878065537948954533115670, −5.82224872228425410715174439825, −4.46350598580106028175366793736, −2.11930968659688568700591720325, 0,
2.11930968659688568700591720325, 4.46350598580106028175366793736, 5.82224872228425410715174439825, 6.37513878065537948954533115670, 7.42021390799642292132016013162, 9.000215228200752369436797457893, 9.907647018896641532019228200236, 10.65492981650156073391598535896, 11.41655744947857510307868720604
