L(s) = 1 | + 3-s − 1.28·5-s − 1.96·7-s + 9-s + 5.51·11-s − 1.28·15-s + 1.82·17-s + 8.08·19-s − 1.96·21-s − 6.97·23-s − 3.34·25-s + 27-s − 2.22·29-s + 4.44·31-s + 5.51·33-s + 2.52·35-s − 2.80·37-s + 4.58·41-s − 5.39·43-s − 1.28·45-s + 5.05·47-s − 3.15·49-s + 1.82·51-s − 2.44·53-s − 7.09·55-s + 8.08·57-s + 10.0·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.575·5-s − 0.740·7-s + 0.333·9-s + 1.66·11-s − 0.332·15-s + 0.442·17-s + 1.85·19-s − 0.427·21-s − 1.45·23-s − 0.668·25-s + 0.192·27-s − 0.413·29-s + 0.798·31-s + 0.960·33-s + 0.426·35-s − 0.460·37-s + 0.715·41-s − 0.822·43-s − 0.191·45-s + 0.737·47-s − 0.450·49-s + 0.255·51-s − 0.336·53-s − 0.956·55-s + 1.07·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.190928210\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.190928210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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
−8.412344210190470361814786580012, −7.70090177960055201273501379276, −7.08325215924006595987597549442, −6.31475971308130082425770434206, −5.57459754808084200368651891667, −4.38836296947183273952851153402, −3.66075719328293990035848963891, −3.26056050412702340053161276777, −1.96476548651816246965508400818, −0.846785388605637742943367256257,
0.846785388605637742943367256257, 1.96476548651816246965508400818, 3.26056050412702340053161276777, 3.66075719328293990035848963891, 4.38836296947183273952851153402, 5.57459754808084200368651891667, 6.31475971308130082425770434206, 7.08325215924006595987597549442, 7.70090177960055201273501379276, 8.412344210190470361814786580012