| L(s) = 1 | − 2·2-s − 7.57·3-s + 4·4-s − 5·5-s + 15.1·6-s − 35.4·7-s − 8·8-s + 30.3·9-s + 10·10-s − 16.6·11-s − 30.2·12-s − 79.9·13-s + 70.8·14-s + 37.8·15-s + 16·16-s − 46.8·17-s − 60.6·18-s − 110.·19-s − 20·20-s + 268.·21-s + 33.2·22-s − 23·23-s + 60.5·24-s + 25·25-s + 159.·26-s − 25.2·27-s − 141.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.45·3-s + 0.5·4-s − 0.447·5-s + 1.03·6-s − 1.91·7-s − 0.353·8-s + 1.12·9-s + 0.316·10-s − 0.455·11-s − 0.728·12-s − 1.70·13-s + 1.35·14-s + 0.651·15-s + 0.250·16-s − 0.667·17-s − 0.794·18-s − 1.33·19-s − 0.223·20-s + 2.78·21-s + 0.322·22-s − 0.208·23-s + 0.515·24-s + 0.200·25-s + 1.20·26-s − 0.180·27-s − 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03207607858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03207607858\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 7.57T + 27T^{2} \) |
| 7 | \( 1 + 35.4T + 343T^{2} \) |
| 11 | \( 1 + 16.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 0.836T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 95.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 331.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 753.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 271.T + 9.12e5T^{2} \) |
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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
−11.70522469191020168940755008056, −10.65060760472449553445011066582, −10.02248467551984072542645536989, −9.092825093184862820472203729348, −7.49205151837231804454471851439, −6.64892521124386147676795191987, −5.89214661485449211643328953521, −4.45471270430794561664189539621, −2.69184090756482823469013641294, −0.14443467134602199433264596732,
0.14443467134602199433264596732, 2.69184090756482823469013641294, 4.45471270430794561664189539621, 5.89214661485449211643328953521, 6.64892521124386147676795191987, 7.49205151837231804454471851439, 9.092825093184862820472203729348, 10.02248467551984072542645536989, 10.65060760472449553445011066582, 11.70522469191020168940755008056
