| L(s) = 1 | + 3.16·2-s − 8.06·3-s + 2.02·4-s + 10.4·5-s − 25.5·6-s − 26.5·7-s − 18.9·8-s + 38.0·9-s + 32.9·10-s + 43.5·11-s − 16.3·12-s − 57.1·13-s − 84.1·14-s − 83.9·15-s − 76.1·16-s − 7.91·17-s + 120.·18-s + 41.9·19-s + 21.0·20-s + 214.·21-s + 137.·22-s + 152.·24-s − 16.7·25-s − 180.·26-s − 89.5·27-s − 53.8·28-s + 231.·29-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 1.55·3-s + 0.253·4-s + 0.930·5-s − 1.73·6-s − 1.43·7-s − 0.835·8-s + 1.41·9-s + 1.04·10-s + 1.19·11-s − 0.393·12-s − 1.21·13-s − 1.60·14-s − 1.44·15-s − 1.18·16-s − 0.112·17-s + 1.57·18-s + 0.506·19-s + 0.235·20-s + 2.22·21-s + 1.33·22-s + 1.29·24-s − 0.134·25-s − 1.36·26-s − 0.638·27-s − 0.363·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.525121995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525121995\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 3.16T + 8T^{2} \) |
| 3 | \( 1 + 8.06T + 27T^{2} \) |
| 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 + 26.5T + 343T^{2} \) |
| 11 | \( 1 - 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.91T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 231.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 372.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 365.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 26.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 136.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 288.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 500.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 181.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 150.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 645.T + 9.12e5T^{2} \) |
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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
−10.42952296332705373989041728143, −9.769919775542855457333187056499, −9.076212454663537235259045096993, −6.92855571360362784605790331932, −6.43095509972032479725114609781, −5.76956528951248260724491425696, −4.98391988214355485173018604778, −3.97073476464021801460665343985, −2.64407816155247160274615332671, −0.67789851657221676017784459693,
0.67789851657221676017784459693, 2.64407816155247160274615332671, 3.97073476464021801460665343985, 4.98391988214355485173018604778, 5.76956528951248260724491425696, 6.43095509972032479725114609781, 6.92855571360362784605790331932, 9.076212454663537235259045096993, 9.769919775542855457333187056499, 10.42952296332705373989041728143
