| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 4.03·5-s + 6·6-s + 17.2·7-s + 8·8-s + 9·9-s − 8.07·10-s − 41.8·11-s + 12·12-s − 0.844·13-s + 34.5·14-s − 12.1·15-s + 16·16-s − 15.4·17-s + 18·18-s − 16.1·20-s + 51.8·21-s − 83.7·22-s + 165.·23-s + 24·24-s − 108.·25-s − 1.68·26-s + 27·27-s + 69.0·28-s − 21.1·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.361·5-s + 0.408·6-s + 0.932·7-s + 0.353·8-s + 0.333·9-s − 0.255·10-s − 1.14·11-s + 0.288·12-s − 0.0180·13-s + 0.659·14-s − 0.208·15-s + 0.250·16-s − 0.220·17-s + 0.235·18-s − 0.180·20-s + 0.538·21-s − 0.812·22-s + 1.49·23-s + 0.204·24-s − 0.869·25-s − 0.0127·26-s + 0.192·27-s + 0.466·28-s − 0.135·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.708532726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.708532726\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 4.03T + 125T^{2} \) |
| 7 | \( 1 - 17.2T + 343T^{2} \) |
| 11 | \( 1 + 41.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.844T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 335.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 535.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 894.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 17.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 639.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.562544905095564738952077484792, −7.70344516043738573666441248203, −7.47796599619461026311459168459, −6.29548602477026722098170915044, −5.31119749962142835060057487995, −4.68797219258736679216634803525, −3.88296471015192006549773607566, −2.83200262357201435560446174327, −2.14888448134179678876249453144, −0.865727060229448452259339101729,
0.865727060229448452259339101729, 2.14888448134179678876249453144, 2.83200262357201435560446174327, 3.88296471015192006549773607566, 4.68797219258736679216634803525, 5.31119749962142835060057487995, 6.29548602477026722098170915044, 7.47796599619461026311459168459, 7.70344516043738573666441248203, 8.562544905095564738952077484792
