| L(s) = 1 | + 3-s + 9-s + 11-s − 2·13-s + 4·17-s − 2·19-s − 5·25-s + 27-s − 4·29-s + 2·31-s + 33-s − 37-s − 2·39-s − 10·41-s − 2·43-s + 8·47-s + 4·51-s + 10·53-s − 2·57-s + 10·61-s − 8·67-s + 16·71-s − 10·73-s − 5·75-s − 14·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.970·17-s − 0.458·19-s − 25-s + 0.192·27-s − 0.742·29-s + 0.359·31-s + 0.174·33-s − 0.164·37-s − 0.320·39-s − 1.56·41-s − 0.304·43-s + 1.16·47-s + 0.560·51-s + 1.37·53-s − 0.264·57-s + 1.28·61-s − 0.977·67-s + 1.89·71-s − 1.17·73-s − 0.577·75-s − 1.57·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239316 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239316 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.05640146177055, −12.80559784141615, −12.04394235025511, −11.87683843199501, −11.42096199070522, −10.65501146011986, −10.24782586420929, −9.857267364493803, −9.477832924701569, −8.833512765628346, −8.464276429694862, −8.020194325468181, −7.348162232604580, −7.185842614449008, −6.523122505104348, −5.902844056177354, −5.443004704597185, −4.958440698052990, −4.193134424241182, −3.868759413191684, −3.308860997697467, −2.700218349517991, −2.100743787617118, −1.608762844020223, −0.8413136725768862, 0,
0.8413136725768862, 1.608762844020223, 2.100743787617118, 2.700218349517991, 3.308860997697467, 3.868759413191684, 4.193134424241182, 4.958440698052990, 5.443004704597185, 5.902844056177354, 6.523122505104348, 7.185842614449008, 7.348162232604580, 8.020194325468181, 8.464276429694862, 8.833512765628346, 9.477832924701569, 9.857267364493803, 10.24782586420929, 10.65501146011986, 11.42096199070522, 11.87683843199501, 12.04394235025511, 12.80559784141615, 13.05640146177055
