| L(s) = 1 | − 2.27·2-s + 3.24·3-s + 3.18·4-s − 3.77·5-s − 7.39·6-s + 3.04·7-s − 2.68·8-s + 7.56·9-s + 8.58·10-s − 11-s + 10.3·12-s + 2.98·13-s − 6.92·14-s − 12.2·15-s − 0.246·16-s + 3.51·17-s − 17.2·18-s + 7.12·19-s − 11.9·20-s + 9.88·21-s + 2.27·22-s − 5.25·23-s − 8.73·24-s + 9.22·25-s − 6.80·26-s + 14.8·27-s + 9.67·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.87·3-s + 1.59·4-s − 1.68·5-s − 3.01·6-s + 1.14·7-s − 0.949·8-s + 2.52·9-s + 2.71·10-s − 0.301·11-s + 2.98·12-s + 0.828·13-s − 1.85·14-s − 3.16·15-s − 0.0615·16-s + 0.852·17-s − 4.05·18-s + 1.63·19-s − 2.68·20-s + 2.15·21-s + 0.485·22-s − 1.09·23-s − 1.78·24-s + 1.84·25-s − 1.33·26-s + 2.85·27-s + 1.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.905689109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905689109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 9.02T + 31T^{2} \) |
| 37 | \( 1 - 0.681T + 37T^{2} \) |
| 41 | \( 1 + 0.935T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 + 0.475T + 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−8.116639020471353298990221533916, −7.70445417910181837352360236256, −7.51290318842315514000431819248, −6.51409561942868567960288653136, −4.86459218866802919436553867100, −4.20288857263136041915436463782, −3.32645429923054388201463898739, −2.75182863196009660946586576723, −1.54530118965232879714439980631, −0.951590683750628320904756416397,
0.951590683750628320904756416397, 1.54530118965232879714439980631, 2.75182863196009660946586576723, 3.32645429923054388201463898739, 4.20288857263136041915436463782, 4.86459218866802919436553867100, 6.51409561942868567960288653136, 7.51290318842315514000431819248, 7.70445417910181837352360236256, 8.116639020471353298990221533916
