| L(s) = 1 | + 2.73·3-s + 5-s + 2·7-s + 4.46·9-s − 3.46·11-s − 2.73·13-s + 2.73·15-s − 3.46·17-s + 19-s + 5.46·21-s − 3.46·23-s + 25-s + 3.99·27-s + 3.46·29-s − 1.46·31-s − 9.46·33-s + 2·35-s + 6.73·37-s − 7.46·39-s − 6·41-s − 4.92·43-s + 4.46·45-s + 12.9·47-s − 3·49-s − 9.46·51-s − 10.7·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.447·5-s + 0.755·7-s + 1.48·9-s − 1.04·11-s − 0.757·13-s + 0.705·15-s − 0.840·17-s + 0.229·19-s + 1.19·21-s − 0.722·23-s + 0.200·25-s + 0.769·27-s + 0.643·29-s − 0.262·31-s − 1.64·33-s + 0.338·35-s + 1.10·37-s − 1.19·39-s − 0.937·41-s − 0.751·43-s + 0.665·45-s + 1.88·47-s − 0.428·49-s − 1.32·51-s − 1.47·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.313081407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.313081407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−11.24458296164480425319556749824, −10.16504772441996179259738013179, −9.483940699053012872537189687096, −8.425170139690295072472280641962, −7.933651046001422676992367948173, −6.91665718516314033151540346617, −5.34029417693935189242495196863, −4.26524606504058423658859734907, −2.81262026871803154044574947246, −2.00908948934280587162148736490,
2.00908948934280587162148736490, 2.81262026871803154044574947246, 4.26524606504058423658859734907, 5.34029417693935189242495196863, 6.91665718516314033151540346617, 7.933651046001422676992367948173, 8.425170139690295072472280641962, 9.483940699053012872537189687096, 10.16504772441996179259738013179, 11.24458296164480425319556749824
