L(s) = 1 | − 2.35·3-s + 5-s − 1.59·7-s + 2.55·9-s − 2.22·11-s + 3.94·13-s − 2.35·15-s + 6.26·17-s + 3.37·19-s + 3.74·21-s + 4.60·23-s + 25-s + 1.04·27-s + 7.90·29-s + 1.30·31-s + 5.23·33-s − 1.59·35-s + 1.20·37-s − 9.28·39-s + 9.85·41-s − 4.13·43-s + 2.55·45-s − 3.91·47-s − 4.47·49-s − 14.7·51-s − 11.1·53-s − 2.22·55-s + ⋯ |
L(s) = 1 | − 1.36·3-s + 0.447·5-s − 0.601·7-s + 0.851·9-s − 0.669·11-s + 1.09·13-s − 0.608·15-s + 1.52·17-s + 0.774·19-s + 0.817·21-s + 0.960·23-s + 0.200·25-s + 0.201·27-s + 1.46·29-s + 0.234·31-s + 0.911·33-s − 0.268·35-s + 0.198·37-s − 1.48·39-s + 1.53·41-s − 0.630·43-s + 0.380·45-s − 0.570·47-s − 0.638·49-s − 2.06·51-s − 1.53·53-s − 0.299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.398827244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.398827244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 6.26T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 - 7.90T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 + 0.0337T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 - 8.62T + 97T^{2} \) |
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
−7.74950910767488627946024218349, −6.92196041851485989790371874335, −6.22092921321723750447411415401, −5.87268278358449273549801157794, −5.16095601241092459582105003541, −4.61356578572279040090535009203, −3.35552404347692130398680487312, −2.88346559059303987264151407047, −1.36549935584738940296749745331, −0.70823955599326522044180717507,
0.70823955599326522044180717507, 1.36549935584738940296749745331, 2.88346559059303987264151407047, 3.35552404347692130398680487312, 4.61356578572279040090535009203, 5.16095601241092459582105003541, 5.87268278358449273549801157794, 6.22092921321723750447411415401, 6.92196041851485989790371874335, 7.74950910767488627946024218349