L(s) = 1 | + 2-s − 3.11·3-s + 4-s + 5-s − 3.11·6-s − 4.50·7-s + 8-s + 6.72·9-s + 10-s − 4.33·11-s − 3.11·12-s − 3.72·13-s − 4.50·14-s − 3.11·15-s + 16-s − 1.11·17-s + 6.72·18-s − 4.50·19-s + 20-s + 14.0·21-s − 4.33·22-s − 3.11·24-s + 25-s − 3.72·26-s − 11.6·27-s − 4.50·28-s − 8.23·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.447·5-s − 1.27·6-s − 1.70·7-s + 0.353·8-s + 2.24·9-s + 0.316·10-s − 1.30·11-s − 0.900·12-s − 1.03·13-s − 1.20·14-s − 0.805·15-s + 0.250·16-s − 0.271·17-s + 1.58·18-s − 1.03·19-s + 0.223·20-s + 3.06·21-s − 0.924·22-s − 0.636·24-s + 0.200·25-s − 0.731·26-s − 2.23·27-s − 0.852·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3758378563\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3758378563\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 - 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 - 0.642T + 97T^{2} \) |
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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
−7.80291975384119026541036630692, −7.05800244529518647377369516779, −6.48352626563470051816625721910, −6.03222030876671724833400775412, −5.31998746870685384338984166895, −4.83721550610330969470009775061, −3.91010174237304585126725868440, −2.88157900574706641779196063538, −1.97621079577623223500824879106, −0.30239563090186820739142845946,
0.30239563090186820739142845946, 1.97621079577623223500824879106, 2.88157900574706641779196063538, 3.91010174237304585126725868440, 4.83721550610330969470009775061, 5.31998746870685384338984166895, 6.03222030876671724833400775412, 6.48352626563470051816625721910, 7.05800244529518647377369516779, 7.80291975384119026541036630692