| L(s) = 1 | − 2.76·2-s − 2.36·3-s + 5.61·4-s + 5-s + 6.53·6-s + 0.315·7-s − 9.99·8-s + 2.59·9-s − 2.76·10-s + 11-s − 13.2·12-s + 6.18·13-s − 0.870·14-s − 2.36·15-s + 16.3·16-s + 17-s − 7.17·18-s + 3.27·19-s + 5.61·20-s − 0.746·21-s − 2.76·22-s − 3.09·23-s + 23.6·24-s + 25-s − 17.0·26-s + 0.951·27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.36·3-s + 2.80·4-s + 0.447·5-s + 2.66·6-s + 0.119·7-s − 3.53·8-s + 0.865·9-s − 0.872·10-s + 0.301·11-s − 3.83·12-s + 1.71·13-s − 0.232·14-s − 0.610·15-s + 4.08·16-s + 0.242·17-s − 1.69·18-s + 0.750·19-s + 1.25·20-s − 0.162·21-s − 0.588·22-s − 0.645·23-s + 4.82·24-s + 0.200·25-s − 3.35·26-s + 0.183·27-s + 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4711435540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4711435540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 2.36T + 3T^{2} \) |
| 7 | \( 1 - 0.315T + 7T^{2} \) |
| 13 | \( 1 - 6.18T + 13T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 - 6.57T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 + 9.93T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 + 4.28T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 - 3.49T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 7.98T + 97T^{2} \) |
| 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
−10.01352568173708999929894484707, −9.382284314647400574625976119237, −8.463882152264207298870237394705, −7.71194366333658557733622089346, −6.48068155779094056269292343370, −6.28072848927549082345987343617, −5.30149498531278204272249648655, −3.39416822601757216976613796615, −1.74598978993772675280405251955, −0.811548916281060897928576118248,
0.811548916281060897928576118248, 1.74598978993772675280405251955, 3.39416822601757216976613796615, 5.30149498531278204272249648655, 6.28072848927549082345987343617, 6.48068155779094056269292343370, 7.71194366333658557733622089346, 8.463882152264207298870237394705, 9.382284314647400574625976119237, 10.01352568173708999929894484707
