| L(s) = 1 | − 0.792·2-s + 3-s − 1.37·4-s + 3.37·5-s − 0.792·6-s − 2.52·7-s + 2.67·8-s + 9-s − 2.67·10-s − 1.37·12-s + 5.84·13-s + 2·14-s + 3.37·15-s + 0.627·16-s − 2.67·17-s − 0.792·18-s + 0.939·19-s − 4.62·20-s − 2.52·21-s + 2·23-s + 2.67·24-s + 6.37·25-s − 4.62·26-s + 27-s + 3.46·28-s + 0.792·29-s − 2.67·30-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + 0.577·3-s − 0.686·4-s + 1.50·5-s − 0.323·6-s − 0.954·7-s + 0.944·8-s + 0.333·9-s − 0.844·10-s − 0.396·12-s + 1.61·13-s + 0.534·14-s + 0.870·15-s + 0.156·16-s − 0.648·17-s − 0.186·18-s + 0.215·19-s − 1.03·20-s − 0.550·21-s + 0.417·23-s + 0.545·24-s + 1.27·25-s − 0.907·26-s + 0.192·27-s + 0.654·28-s + 0.147·29-s − 0.487·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296056224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296056224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 0.939T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 0.792T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.15T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 0.627T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.00996639529494700746195103324, −10.20276855130736505277057435377, −9.315621830151822790844752701922, −9.081872893525148840841880675454, −7.965966866549578188069293704014, −6.55905068429845032387359770143, −5.79669009434291422738909357651, −4.34392674455571782839217239249, −2.97522498873783977257293447587, −1.40036546844204146468030674313,
1.40036546844204146468030674313, 2.97522498873783977257293447587, 4.34392674455571782839217239249, 5.79669009434291422738909357651, 6.55905068429845032387359770143, 7.965966866549578188069293704014, 9.081872893525148840841880675454, 9.315621830151822790844752701922, 10.20276855130736505277057435377, 11.00996639529494700746195103324
