| L(s) = 1 | + 5.90e4·3-s + 1.51e7·5-s − 8.40e8·7-s + 3.48e9·9-s − 3.34e10·11-s − 3.01e11·13-s + 8.93e11·15-s + 5.14e12·17-s + 4.56e13·19-s − 4.96e13·21-s + 2.41e13·23-s − 2.47e14·25-s + 2.05e14·27-s + 7.82e14·29-s + 8.00e15·31-s − 1.97e15·33-s − 1.27e16·35-s + 2.06e16·37-s − 1.77e16·39-s − 7.03e16·41-s + 6.95e15·43-s + 5.27e16·45-s − 2.12e17·47-s + 1.48e17·49-s + 3.03e17·51-s + 8.69e17·53-s − 5.06e17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.693·5-s − 1.12·7-s + 0.333·9-s − 0.389·11-s − 0.605·13-s + 0.400·15-s + 0.618·17-s + 1.70·19-s − 0.649·21-s + 0.121·23-s − 0.519·25-s + 0.192·27-s + 0.345·29-s + 1.75·31-s − 0.224·33-s − 0.779·35-s + 0.706·37-s − 0.349·39-s − 0.818·41-s + 0.0490·43-s + 0.231·45-s − 0.589·47-s + 0.265·49-s + 0.357·51-s + 0.682·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.678631510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.678631510\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
| good | 5 | \( 1 - 1.51e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 8.40e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.34e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.01e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 5.14e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.56e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.41e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 7.82e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 8.00e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.06e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.03e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 6.95e15T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.12e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 8.69e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 9.80e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 5.62e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.36e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.99e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.63e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.05e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 8.96e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.71e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 3.93e20T + 5.27e41T^{2} \) |
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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
−13.24946089075749263368831497600, −12.00203764494187239692616813898, −10.02068701766496791631639238167, −9.552997313669869539024598391341, −7.906190898629386068097696633591, −6.56709444262571735979837336660, −5.20991534354749667434179922198, −3.40550180442753395537849298506, −2.43850099094929870349059792233, −0.844104728546681090579063769177,
0.844104728546681090579063769177, 2.43850099094929870349059792233, 3.40550180442753395537849298506, 5.20991534354749667434179922198, 6.56709444262571735979837336660, 7.906190898629386068097696633591, 9.552997313669869539024598391341, 10.02068701766496791631639238167, 12.00203764494187239692616813898, 13.24946089075749263368831497600
