L(s) = 1 | + 1.37·3-s − 0.949·5-s + 7-s − 1.11·9-s − 11-s + 13-s − 1.30·15-s − 5.27·17-s − 3.79·19-s + 1.37·21-s + 2.13·23-s − 4.09·25-s − 5.65·27-s + 2.06·29-s − 0.698·31-s − 1.37·33-s − 0.949·35-s + 5.10·37-s + 1.37·39-s − 1.72·41-s + 0.353·43-s + 1.05·45-s + 12.2·47-s + 49-s − 7.24·51-s + 12.3·53-s + 0.949·55-s + ⋯ |
L(s) = 1 | + 0.793·3-s − 0.424·5-s + 0.377·7-s − 0.370·9-s − 0.301·11-s + 0.277·13-s − 0.336·15-s − 1.27·17-s − 0.870·19-s + 0.299·21-s + 0.444·23-s − 0.819·25-s − 1.08·27-s + 0.384·29-s − 0.125·31-s − 0.239·33-s − 0.160·35-s + 0.839·37-s + 0.220·39-s − 0.269·41-s + 0.0539·43-s + 0.157·45-s + 1.78·47-s + 0.142·49-s − 1.01·51-s + 1.69·53-s + 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.005778425\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.005778425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.37T + 3T^{2} \) |
| 5 | \( 1 + 0.949T + 5T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 + 0.698T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 0.353T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 0.154T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 - 3.90T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 6.64T + 89T^{2} \) |
| 97 | \( 1 - 9.20T + 97T^{2} \) |
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
−8.024251377695937305420851057265, −7.28076690768276072564276295449, −6.52684825024526651781952790323, −5.77309219367989877838028403322, −4.93822659145617521618527271394, −4.10867144212817016210287402082, −3.61095894675585764311596158876, −2.46554655785580945253011285677, −2.15164517450064558272704458266, −0.64519794628026040364954156739,
0.64519794628026040364954156739, 2.15164517450064558272704458266, 2.46554655785580945253011285677, 3.61095894675585764311596158876, 4.10867144212817016210287402082, 4.93822659145617521618527271394, 5.77309219367989877838028403322, 6.52684825024526651781952790323, 7.28076690768276072564276295449, 8.024251377695937305420851057265