| L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s + 5·11-s − 13-s + 3·15-s − 17-s − 3·19-s − 21-s − 5·23-s + 4·25-s + 27-s + 2·29-s + 5·33-s − 3·35-s − 2·37-s − 39-s + 7·41-s − 3·43-s + 3·45-s + 12·47-s + 49-s − 51-s − 8·53-s + 15·55-s − 3·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s − 0.688·19-s − 0.218·21-s − 1.04·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s + 0.870·33-s − 0.507·35-s − 0.328·37-s − 0.160·39-s + 1.09·41-s − 0.457·43-s + 0.447·45-s + 1.75·47-s + 1/7·49-s − 0.140·51-s − 1.09·53-s + 2.02·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.054524015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.054524015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.49843059955769, −14.72012021690504, −14.28105283441123, −13.98661233722294, −13.47372971928918, −12.81363939328449, −12.37076210070121, −11.78749265611220, −11.02084479207643, −10.36762019643803, −9.836015744506314, −9.448222347339062, −8.915513795267085, −8.471653548441130, −7.583678680881843, −6.937104404765398, −6.224049534205340, −6.105581156641971, −5.183196509724414, −4.319109148517318, −3.858666966566378, −2.989134543959161, −2.171083853879936, −1.785714801761716, −0.7869294560334914,
0.7869294560334914, 1.785714801761716, 2.171083853879936, 2.989134543959161, 3.858666966566378, 4.319109148517318, 5.183196509724414, 6.105581156641971, 6.224049534205340, 6.937104404765398, 7.583678680881843, 8.471653548441130, 8.915513795267085, 9.448222347339062, 9.836015744506314, 10.36762019643803, 11.02084479207643, 11.78749265611220, 12.37076210070121, 12.81363939328449, 13.47372971928918, 13.98661233722294, 14.28105283441123, 14.72012021690504, 15.49843059955769
