L(s) = 1 | − 1.22·3-s − 3.08·7-s − 1.50·9-s + 3.83·11-s + 5.85·13-s + 6.54·17-s − 19-s + 3.77·21-s − 4.37·23-s + 5.50·27-s − 4.41·29-s − 0.451·31-s − 4.69·33-s + 2.49·37-s − 7.15·39-s − 1.03·41-s − 3.19·43-s + 3.70·47-s + 2.52·49-s − 7.99·51-s + 5.19·53-s + 1.22·57-s + 7.18·59-s + 5.02·61-s + 4.65·63-s + 13.1·67-s + 5.34·69-s + ⋯ |
L(s) = 1 | − 0.705·3-s − 1.16·7-s − 0.502·9-s + 1.15·11-s + 1.62·13-s + 1.58·17-s − 0.229·19-s + 0.822·21-s − 0.911·23-s + 1.05·27-s − 0.820·29-s − 0.0810·31-s − 0.816·33-s + 0.409·37-s − 1.14·39-s − 0.162·41-s − 0.486·43-s + 0.540·47-s + 0.360·49-s − 1.11·51-s + 0.713·53-s + 0.161·57-s + 0.935·59-s + 0.643·61-s + 0.586·63-s + 1.60·67-s + 0.642·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352427485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352427485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 0.451T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 0.245T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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
−7.906182775249490252928387461164, −6.91390035204496122069974394612, −6.37713725128047456125284768127, −5.83348601316420133184014816230, −5.43391112859017629860520220214, −4.04478270902447417171288729804, −3.67199511546127179104192265454, −2.89066976620685150979964910531, −1.53427141707878578373515680252, −0.63766714972393195599121083970,
0.63766714972393195599121083970, 1.53427141707878578373515680252, 2.89066976620685150979964910531, 3.67199511546127179104192265454, 4.04478270902447417171288729804, 5.43391112859017629860520220214, 5.83348601316420133184014816230, 6.37713725128047456125284768127, 6.91390035204496122069974394612, 7.906182775249490252928387461164