| L(s) = 1 | + 0.289·2-s + 3-s − 1.91·4-s + 0.289·6-s − 4.91·7-s − 1.13·8-s + 9-s + 4.91·11-s − 1.91·12-s − 13-s − 1.42·14-s + 3.50·16-s + 4.33·17-s + 0.289·18-s + 2.57·19-s − 4.91·21-s + 1.42·22-s + 6.33·23-s − 1.13·24-s − 0.289·26-s + 27-s + 9.42·28-s + 6·29-s + 1.42·31-s + 3.27·32-s + 4.91·33-s + 1.25·34-s + ⋯ |
| L(s) = 1 | + 0.204·2-s + 0.577·3-s − 0.958·4-s + 0.118·6-s − 1.85·7-s − 0.400·8-s + 0.333·9-s + 1.48·11-s − 0.553·12-s − 0.277·13-s − 0.379·14-s + 0.876·16-s + 1.05·17-s + 0.0681·18-s + 0.591·19-s − 1.07·21-s + 0.303·22-s + 1.32·23-s − 0.231·24-s − 0.0567·26-s + 0.192·27-s + 1.78·28-s + 1.11·29-s + 0.255·31-s + 0.579·32-s + 0.855·33-s + 0.215·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520610094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520610094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.289T + 2T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 0.338T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 - 0.916T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 0.338T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.713398423069702661562726275913, −9.248032153632190792518895772745, −8.686834208183750458505261644683, −7.40341131345381536583039688956, −6.60869261024382651076247522709, −5.73621443665805365788928110018, −4.53489090238216368852424613228, −3.49062932289989177217503348973, −3.08005417038692581858456229087, −0.953026695536530234843631952437,
0.953026695536530234843631952437, 3.08005417038692581858456229087, 3.49062932289989177217503348973, 4.53489090238216368852424613228, 5.73621443665805365788928110018, 6.60869261024382651076247522709, 7.40341131345381536583039688956, 8.686834208183750458505261644683, 9.248032153632190792518895772745, 9.713398423069702661562726275913
