| L(s) = 1 | + 1.53·2-s + 3-s + 1.34·4-s + 1.53·6-s − 0.347·7-s + 0.532·8-s + 9-s + 1.87·11-s + 1.34·12-s − 1.53·13-s − 0.532·14-s − 0.532·16-s − 1.87·17-s + 1.53·18-s − 0.347·21-s + 2.87·22-s + 0.532·24-s − 2.34·26-s + 27-s − 0.467·28-s − 29-s − 1.34·32-s + 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-s + 41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 3-s + 1.34·4-s + 1.53·6-s − 0.347·7-s + 0.532·8-s + 9-s + 1.87·11-s + 1.34·12-s − 1.53·13-s − 0.532·14-s − 0.532·16-s − 1.87·17-s + 1.53·18-s − 0.347·21-s + 2.87·22-s + 0.532·24-s − 2.34·26-s + 27-s − 0.467·28-s − 29-s − 1.34·32-s + 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.457040280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457040280\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 + 0.347T + T^{2} \) |
| 11 | \( 1 - 1.87T + T^{2} \) |
| 13 | \( 1 + 1.53T + T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.347T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.53T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.310575308123055144816157607234, −8.606309422158384492736987840695, −7.31224669850524849697821884818, −6.82883412006812277965661625110, −6.17209459992901541821773673578, −4.93368819348352006813103738464, −4.23692971573057199033517471435, −3.72530078415204495070506315368, −2.67132526055191886201146298813, −1.94074726955905458397913084652,
1.94074726955905458397913084652, 2.67132526055191886201146298813, 3.72530078415204495070506315368, 4.23692971573057199033517471435, 4.93368819348352006813103738464, 6.17209459992901541821773673578, 6.82883412006812277965661625110, 7.31224669850524849697821884818, 8.606309422158384492736987840695, 9.310575308123055144816157607234
