L(s) = 1 | − 2.31·2-s + 2.43·3-s + 3.33·4-s + 5-s − 5.62·6-s − 3.09·8-s + 2.92·9-s − 2.31·10-s − 11-s + 8.12·12-s + 3.20·13-s + 2.43·15-s + 0.468·16-s + 6.54·17-s − 6.75·18-s + 5.18·19-s + 3.33·20-s + 2.31·22-s − 0.628·23-s − 7.52·24-s + 25-s − 7.40·26-s − 0.188·27-s + 8.12·29-s − 5.62·30-s + 3.17·31-s + 5.10·32-s + ⋯ |
L(s) = 1 | − 1.63·2-s + 1.40·3-s + 1.66·4-s + 0.447·5-s − 2.29·6-s − 1.09·8-s + 0.974·9-s − 0.730·10-s − 0.301·11-s + 2.34·12-s + 0.889·13-s + 0.628·15-s + 0.117·16-s + 1.58·17-s − 1.59·18-s + 1.18·19-s + 0.746·20-s + 0.492·22-s − 0.131·23-s − 1.53·24-s + 0.200·25-s − 1.45·26-s − 0.0363·27-s + 1.50·29-s − 1.02·30-s + 0.569·31-s + 0.902·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.696551842\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696551842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 + 0.628T + 23T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 6.05T + 59T^{2} \) |
| 61 | \( 1 + 0.349T + 61T^{2} \) |
| 67 | \( 1 - 2.18T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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
−8.678104670508045728503766978531, −8.368359878688985637796941830902, −7.67805439173650309786917028201, −7.03978536907374074992604646453, −6.06580233977391592984575829073, −4.99667578744171461473264587142, −3.48010228315342660022175508496, −2.94379129351713899393564888335, −1.84406875703634420617952453683, −1.05267491498905046360166366469,
1.05267491498905046360166366469, 1.84406875703634420617952453683, 2.94379129351713899393564888335, 3.48010228315342660022175508496, 4.99667578744171461473264587142, 6.06580233977391592984575829073, 7.03978536907374074992604646453, 7.67805439173650309786917028201, 8.368359878688985637796941830902, 8.678104670508045728503766978531