| L(s) = 1 | − 3.84·2-s + 3·3-s + 6.78·4-s − 11.5·6-s − 25.3·7-s + 4.68·8-s + 9·9-s − 56.8·11-s + 20.3·12-s − 4.98·13-s + 97.6·14-s − 72.2·16-s − 129.·17-s − 34.6·18-s + 24.2·19-s − 76.1·21-s + 218.·22-s − 81.3·23-s + 14.0·24-s + 19.1·26-s + 27·27-s − 172.·28-s − 112.·29-s − 245.·31-s + 240.·32-s − 170.·33-s + 498.·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.577·3-s + 0.847·4-s − 0.784·6-s − 1.37·7-s + 0.206·8-s + 0.333·9-s − 1.55·11-s + 0.489·12-s − 0.106·13-s + 1.86·14-s − 1.12·16-s − 1.84·17-s − 0.453·18-s + 0.292·19-s − 0.791·21-s + 2.11·22-s − 0.737·23-s + 0.119·24-s + 0.144·26-s + 0.192·27-s − 1.16·28-s − 0.722·29-s − 1.42·31-s + 1.32·32-s − 0.900·33-s + 2.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02108854588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02108854588\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.84T + 8T^{2} \) |
| 7 | \( 1 + 25.3T + 343T^{2} \) |
| 11 | \( 1 + 56.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.98T + 2.19e3T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 377.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 656.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 765.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 115.T + 9.12e5T^{2} \) |
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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
−8.828947928436027711859599787958, −8.366368105495417140991069046887, −7.34997874213053402451550356284, −6.98841463883247691436642592263, −5.90091746171820711523376563112, −4.75590210515232316514367745627, −3.63022760711902220773881142262, −2.60248013724920657855916756529, −1.85101020484814045327414792714, −0.07559028042246165474671089029,
0.07559028042246165474671089029, 1.85101020484814045327414792714, 2.60248013724920657855916756529, 3.63022760711902220773881142262, 4.75590210515232316514367745627, 5.90091746171820711523376563112, 6.98841463883247691436642592263, 7.34997874213053402451550356284, 8.366368105495417140991069046887, 8.828947928436027711859599787958
