L(s) = 1 | − 4.84e5·3-s + 2.23e7·5-s − 2.34e9·7-s + 1.40e11·9-s + 7.12e11·11-s − 7.52e12·13-s − 1.08e13·15-s + 8.04e13·17-s + 1.95e13·19-s + 1.13e15·21-s − 2.11e15·23-s − 1.14e16·25-s − 2.22e16·27-s − 1.03e17·29-s + 1.45e17·31-s − 3.44e17·33-s − 5.25e16·35-s + 3.23e17·37-s + 3.64e18·39-s + 4.65e18·41-s − 1.07e19·43-s + 3.13e18·45-s + 2.32e19·47-s − 2.18e19·49-s − 3.89e19·51-s − 2.42e19·53-s + 1.59e19·55-s + ⋯ |
L(s) = 1 | − 1.57·3-s + 0.205·5-s − 0.448·7-s + 1.48·9-s + 0.753·11-s − 1.16·13-s − 0.323·15-s + 0.569·17-s + 0.0385·19-s + 0.707·21-s − 0.463·23-s − 0.957·25-s − 0.771·27-s − 1.58·29-s + 1.02·31-s − 1.18·33-s − 0.0919·35-s + 0.299·37-s + 1.83·39-s + 1.32·41-s − 1.75·43-s + 0.305·45-s + 1.37·47-s − 0.798·49-s − 0.898·51-s − 0.359·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{25}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 4.84e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 2.23e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 2.34e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 7.12e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 7.52e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 8.04e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 1.95e13T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.11e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.03e17T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.45e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 3.23e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.65e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.07e19T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.32e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 2.42e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.61e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.11e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.38e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.11e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.17e22T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.34e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.06e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 8.47e22T + 4.96e45T^{2} \) |
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
−10.14285284355095749173948058106, −9.477588731398450749392124759675, −7.66852109473761274078113721531, −6.59385279730155005563299099735, −5.81280945655987919612146817899, −4.89087032948316014184778720626, −3.73421776636622768755036673355, −2.11713134371423374934703315967, −0.877913228866284754695563259813, 0,
0.877913228866284754695563259813, 2.11713134371423374934703315967, 3.73421776636622768755036673355, 4.89087032948316014184778720626, 5.81280945655987919612146817899, 6.59385279730155005563299099735, 7.66852109473761274078113721531, 9.477588731398450749392124759675, 10.14285284355095749173948058106