| L(s) = 1 | − 1.57e5·3-s + 3.23e7·5-s + 8.65e8·7-s + 1.44e10·9-s − 3.34e10·11-s + 6.73e11·13-s − 5.10e12·15-s − 9.68e12·17-s − 2.47e13·19-s − 1.36e14·21-s + 2.38e14·23-s + 5.68e14·25-s − 6.26e14·27-s + 1.28e15·29-s + 5.18e14·31-s + 5.27e15·33-s + 2.79e16·35-s − 4.73e16·37-s − 1.06e17·39-s + 5.43e16·41-s + 1.52e17·43-s + 4.66e17·45-s + 4.11e17·47-s + 1.90e17·49-s + 1.52e18·51-s − 5.65e17·53-s − 1.08e18·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s + 1.48·5-s + 1.15·7-s + 1.37·9-s − 0.388·11-s + 1.35·13-s − 2.28·15-s − 1.16·17-s − 0.927·19-s − 1.78·21-s + 1.19·23-s + 1.19·25-s − 0.585·27-s + 0.565·29-s + 0.113·31-s + 0.599·33-s + 1.71·35-s − 1.61·37-s − 2.09·39-s + 0.632·41-s + 1.07·43-s + 2.04·45-s + 1.14·47-s + 0.341·49-s + 1.79·51-s − 0.444·53-s − 0.575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.885681674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885681674\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 1.57e5T + 1.04e10T^{2} \) |
| 5 | \( 1 - 3.23e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 8.65e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.34e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.73e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.68e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.47e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.38e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.28e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.18e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.73e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 5.43e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.52e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.11e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 5.65e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.43e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 5.28e17T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.27e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.85e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 4.09e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 4.81e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.70e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.94e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.01e21T + 5.27e41T^{2} \) |
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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
−13.92799269276517091347048616049, −12.78341607833579334347265005435, −11.11488889762732775232194085400, −10.61046284675042703354247562155, −8.789294081899059015663120161713, −6.62447223698133790380739753415, −5.67199427653631600709075061768, −4.67810357656093182468070626225, −2.00646242669092432633792706921, −0.903483395397375090454236209379,
0.903483395397375090454236209379, 2.00646242669092432633792706921, 4.67810357656093182468070626225, 5.67199427653631600709075061768, 6.62447223698133790380739753415, 8.789294081899059015663120161713, 10.61046284675042703354247562155, 11.11488889762732775232194085400, 12.78341607833579334347265005435, 13.92799269276517091347048616049
