| L(s) = 1 | + 1.56·2-s − 1.66·3-s + 0.460·4-s − 5-s − 2.60·6-s − 3.88·7-s − 2.41·8-s − 0.235·9-s − 1.56·10-s − 0.323·11-s − 0.765·12-s + 1.46·13-s − 6.09·14-s + 1.66·15-s − 4.70·16-s − 6.47·17-s − 0.368·18-s + 2.72·19-s − 0.460·20-s + 6.46·21-s − 0.507·22-s + 6.44·23-s + 4.01·24-s + 25-s + 2.29·26-s + 5.37·27-s − 1.78·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.960·3-s + 0.230·4-s − 0.447·5-s − 1.06·6-s − 1.46·7-s − 0.853·8-s − 0.0783·9-s − 0.496·10-s − 0.0975·11-s − 0.220·12-s + 0.405·13-s − 1.62·14-s + 0.429·15-s − 1.17·16-s − 1.57·17-s − 0.0869·18-s + 0.624·19-s − 0.102·20-s + 1.41·21-s − 0.108·22-s + 1.34·23-s + 0.819·24-s + 0.200·25-s + 0.449·26-s + 1.03·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9250411523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9250411523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 0.323T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 0.164T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 + 8.36T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + 7.43T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
| 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
−9.098898113234671668669277905343, −8.561498967390009990128910243671, −7.09457893093963084600915857870, −6.44202958447358445667900257568, −6.04296444540660670405539829720, −5.01015555427803547499601911825, −4.46132402376645185052099902268, −3.34218734041242561216147850330, −2.80225592618616009774620397005, −0.54116725934959732211041216014,
0.54116725934959732211041216014, 2.80225592618616009774620397005, 3.34218734041242561216147850330, 4.46132402376645185052099902268, 5.01015555427803547499601911825, 6.04296444540660670405539829720, 6.44202958447358445667900257568, 7.09457893093963084600915857870, 8.561498967390009990128910243671, 9.098898113234671668669277905343
