L(s) = 1 | − 2·2-s + 4·4-s − 15.0·5-s − 7·7-s − 8·8-s + 30.1·10-s + 9.90·11-s − 30.5·13-s + 14·14-s + 16·16-s + 12.8·17-s + 19·19-s − 60.2·20-s − 19.8·22-s − 141.·23-s + 101.·25-s + 61.0·26-s − 28·28-s + 268.·29-s − 95.9·31-s − 32·32-s − 25.6·34-s + 105.·35-s + 242.·37-s − 38·38-s + 120.·40-s − 389.·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.952·10-s + 0.271·11-s − 0.651·13-s + 0.267·14-s + 0.250·16-s + 0.182·17-s + 0.229·19-s − 0.673·20-s − 0.191·22-s − 1.28·23-s + 0.813·25-s + 0.460·26-s − 0.188·28-s + 1.71·29-s − 0.556·31-s − 0.176·32-s − 0.129·34-s + 0.508·35-s + 1.07·37-s − 0.162·38-s + 0.476·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 - 19T \) |
good | 5 | \( 1 + 15.0T + 125T^{2} \) |
| 11 | \( 1 - 9.90T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 389.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 44.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 723.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 277.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 204.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 156.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 670.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 668.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 900.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 899.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.62e3T + 9.12e5T^{2} \) |
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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
−8.065577023804415353199430299670, −7.72497286892125356782058301150, −6.85731081443223207182340589069, −6.16362710217587124975477296673, −4.97309393973847295710709162900, −4.06069378905801581029647415286, −3.29094895717540181670582969148, −2.27177108783212446830401068475, −0.887012697875567199110005922777, 0,
0.887012697875567199110005922777, 2.27177108783212446830401068475, 3.29094895717540181670582969148, 4.06069378905801581029647415286, 4.97309393973847295710709162900, 6.16362710217587124975477296673, 6.85731081443223207182340589069, 7.72497286892125356782058301150, 8.065577023804415353199430299670