| L(s) = 1 | − 70.8·3-s + 2.78e3·5-s + 2.40e3·7-s − 1.46e4·9-s − 5.04e4·11-s + 5.21e4·13-s − 1.97e5·15-s − 2.79e5·17-s − 7.99e5·19-s − 1.70e5·21-s − 1.91e6·23-s + 5.82e6·25-s + 2.43e6·27-s − 3.59e6·29-s + 3.40e6·31-s + 3.57e6·33-s + 6.69e6·35-s + 7.70e6·37-s − 3.69e6·39-s − 1.66e7·41-s + 5.41e6·43-s − 4.08e7·45-s + 1.84e7·47-s + 5.76e6·49-s + 1.97e7·51-s − 8.32e7·53-s − 1.40e8·55-s + ⋯ |
| L(s) = 1 | − 0.505·3-s + 1.99·5-s + 0.377·7-s − 0.744·9-s − 1.03·11-s + 0.506·13-s − 1.00·15-s − 0.811·17-s − 1.40·19-s − 0.190·21-s − 1.43·23-s + 2.98·25-s + 0.881·27-s − 0.944·29-s + 0.662·31-s + 0.524·33-s + 0.754·35-s + 0.675·37-s − 0.255·39-s − 0.917·41-s + 0.241·43-s − 1.48·45-s + 0.551·47-s + 0.142·49-s + 0.409·51-s − 1.45·53-s − 2.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 + 70.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.78e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 5.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.21e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.99e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.91e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.59e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.40e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.66e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.41e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.84e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.45e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.82e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.88e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.56e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.93e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.82e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.99193285055328845762613989646, −10.44281673173423532926156087770, −9.267797019475163818678256625811, −8.227897422334648374966341037810, −6.35836500095478997134958770491, −5.84566678903723056886336692934, −4.76342857029709052640502079722, −2.60089364871424124587013638431, −1.72745791280549328105950550392, 0,
1.72745791280549328105950550392, 2.60089364871424124587013638431, 4.76342857029709052640502079722, 5.84566678903723056886336692934, 6.35836500095478997134958770491, 8.227897422334648374966341037810, 9.267797019475163818678256625811, 10.44281673173423532926156087770, 10.99193285055328845762613989646
