| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 4·11-s + 2·15-s − 4·17-s + 4·19-s + 4·21-s − 4·23-s − 25-s − 27-s + 2·29-s + 4·31-s − 4·33-s + 8·35-s + 12·37-s + 12·41-s + 8·43-s − 2·45-s + 9·49-s + 4·51-s − 14·53-s − 8·55-s − 4·57-s − 2·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.516·15-s − 0.970·17-s + 0.917·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.696·33-s + 1.35·35-s + 1.97·37-s + 1.87·41-s + 1.21·43-s − 0.298·45-s + 9/7·49-s + 0.560·51-s − 1.92·53-s − 1.07·55-s − 0.529·57-s − 0.260·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.010778271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010778271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.36480626408893, −14.35742842636814, −14.13638869833757, −13.37752206169415, −12.65127944255794, −12.56244969370149, −11.83304005921994, −11.25826177574480, −11.14655884614372, −10.06932050578296, −9.692397481193162, −9.279921014947617, −8.651154341437141, −7.730560320621184, −7.461706726121303, −6.622553524721049, −6.195309235690079, −5.944082430222096, −4.808960579453384, −4.137764385432075, −3.889901875816166, −3.048010053625849, −2.378471589439829, −1.133371373225198, −0.4582134095976856,
0.4582134095976856, 1.133371373225198, 2.378471589439829, 3.048010053625849, 3.889901875816166, 4.137764385432075, 4.808960579453384, 5.944082430222096, 6.195309235690079, 6.622553524721049, 7.461706726121303, 7.730560320621184, 8.651154341437141, 9.279921014947617, 9.692397481193162, 10.06932050578296, 11.14655884614372, 11.25826177574480, 11.83304005921994, 12.56244969370149, 12.65127944255794, 13.37752206169415, 14.13638869833757, 14.35742842636814, 15.36480626408893
