L(s) = 1 | − 3.33·2-s + 3.14·4-s + 7·7-s + 16.2·8-s − 18.3·11-s + 10.1·13-s − 23.3·14-s − 79.2·16-s + 24.6·17-s + 77.4·19-s + 61.1·22-s − 149.·23-s − 33.8·26-s + 21.9·28-s + 10.2·29-s + 124.·31-s + 134.·32-s − 82.4·34-s + 215.·37-s − 258.·38-s − 495.·41-s − 220.·43-s − 57.5·44-s + 498.·46-s − 212.·47-s + 49·49-s + 31.8·52-s + ⋯ |
L(s) = 1 | − 1.18·2-s + 0.392·4-s + 0.377·7-s + 0.716·8-s − 0.502·11-s + 0.216·13-s − 0.446·14-s − 1.23·16-s + 0.352·17-s + 0.934·19-s + 0.592·22-s − 1.35·23-s − 0.255·26-s + 0.148·28-s + 0.0659·29-s + 0.720·31-s + 0.744·32-s − 0.415·34-s + 0.959·37-s − 1.10·38-s − 1.88·41-s − 0.782·43-s − 0.197·44-s + 1.59·46-s − 0.660·47-s + 0.142·49-s + 0.0849·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 + 3.33T + 8T^{2} \) |
| 11 | \( 1 + 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 495.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 220.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 324.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 819.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 12.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 33.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 810.T + 9.12e5T^{2} \) |
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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
−8.431302409552844887048218092715, −8.155822130649367302019163188763, −7.38290793853055917085704931505, −6.44747609415606882227317165859, −5.34986597705328759951354143749, −4.54755104117758441602481240982, −3.38533794506503599883149516020, −2.09112442625158890914985815567, −1.13210883522260627177943633406, 0,
1.13210883522260627177943633406, 2.09112442625158890914985815567, 3.38533794506503599883149516020, 4.54755104117758441602481240982, 5.34986597705328759951354143749, 6.44747609415606882227317165859, 7.38290793853055917085704931505, 8.155822130649367302019163188763, 8.431302409552844887048218092715