| L(s) = 1 | + 3.13·2-s + 3·3-s + 1.80·4-s − 16.9·5-s + 9.39·6-s − 16.5·7-s − 19.3·8-s + 9·9-s − 52.9·10-s − 8.78·11-s + 5.42·12-s − 79.3·13-s − 51.8·14-s − 50.7·15-s − 75.1·16-s − 87.0·17-s + 28.1·18-s + 27.8·19-s − 30.5·20-s − 49.6·21-s − 27.5·22-s + 23·23-s − 58.1·24-s + 160.·25-s − 248.·26-s + 27·27-s − 29.9·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.225·4-s − 1.51·5-s + 0.639·6-s − 0.894·7-s − 0.857·8-s + 0.333·9-s − 1.67·10-s − 0.240·11-s + 0.130·12-s − 1.69·13-s − 0.989·14-s − 0.873·15-s − 1.17·16-s − 1.24·17-s + 0.369·18-s + 0.336·19-s − 0.341·20-s − 0.516·21-s − 0.266·22-s + 0.208·23-s − 0.494·24-s + 1.28·25-s − 1.87·26-s + 0.192·27-s − 0.201·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8688602939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8688602939\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.13T + 8T^{2} \) |
| 5 | \( 1 + 16.9T + 125T^{2} \) |
| 7 | \( 1 + 16.5T + 343T^{2} \) |
| 11 | \( 1 + 8.78T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.8T + 6.85e3T^{2} \) |
| 31 | \( 1 - 235.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 714.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 385.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 689.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 450.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 765.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 843.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.754148011540825215629107443964, −7.942394710484309283336643723869, −7.09071717257021718150598932315, −6.58714631872631015738328078615, −5.25624830192467414011062015810, −4.54982055608954534675499373731, −3.92459673371092315117197176047, −3.09238942431989730073849590124, −2.47597476744581819232322807403, −0.32611542157685553424455689540,
0.32611542157685553424455689540, 2.47597476744581819232322807403, 3.09238942431989730073849590124, 3.92459673371092315117197176047, 4.54982055608954534675499373731, 5.25624830192467414011062015810, 6.58714631872631015738328078615, 7.09071717257021718150598932315, 7.942394710484309283336643723869, 8.754148011540825215629107443964
