| L(s) = 1 | − 0.534·3-s − 2.85·5-s + 1.52·7-s − 2.71·9-s − 3.72·11-s + 1.48·13-s + 1.52·15-s − 6.09·17-s − 8.39·19-s − 0.814·21-s + 3.14·25-s + 3.05·27-s − 1.98·29-s + 3.69·31-s + 1.99·33-s − 4.35·35-s + 9.05·37-s − 0.796·39-s − 2.86·41-s − 2.13·43-s + 7.74·45-s − 9.05·47-s − 4.67·49-s + 3.25·51-s − 1.56·53-s + 10.6·55-s + 4.48·57-s + ⋯ |
| L(s) = 1 | − 0.308·3-s − 1.27·5-s + 0.576·7-s − 0.904·9-s − 1.12·11-s + 0.413·13-s + 0.393·15-s − 1.47·17-s − 1.92·19-s − 0.177·21-s + 0.628·25-s + 0.587·27-s − 0.368·29-s + 0.663·31-s + 0.346·33-s − 0.735·35-s + 1.48·37-s − 0.127·39-s − 0.447·41-s − 0.325·43-s + 1.15·45-s − 1.32·47-s − 0.667·49-s + 0.455·51-s − 0.214·53-s + 1.43·55-s + 0.594·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4536480336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4536480336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.534T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 8.39T + 19T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 + 9.05T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 8.98T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 + 0.584T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.292067273722406067341986077925, −7.979440002997024137466128553248, −6.87450615510749911871597541465, −6.29758181005209570926489400186, −5.32938867950546817942070303508, −4.55677398685920090262293048017, −4.02385342185696780787384671673, −2.89653401956964181512803744371, −2.07835716044246871713284235322, −0.36304465069762870646774909803,
0.36304465069762870646774909803, 2.07835716044246871713284235322, 2.89653401956964181512803744371, 4.02385342185696780787384671673, 4.55677398685920090262293048017, 5.32938867950546817942070303508, 6.29758181005209570926489400186, 6.87450615510749911871597541465, 7.979440002997024137466128553248, 8.292067273722406067341986077925
