| L(s) = 1 | − 3.14·3-s − 5-s + 3.89·7-s + 6.89·9-s + 4.29·11-s + 4.34·13-s + 3.14·15-s + 1.60·17-s + 1.20·19-s − 12.2·21-s + 8.34·23-s + 25-s − 12.2·27-s − 29-s + 2.39·31-s − 13.4·33-s − 3.89·35-s − 9.78·37-s − 13.6·39-s + 5.78·41-s − 3.60·43-s − 6.89·45-s − 8.58·47-s + 8.14·49-s − 5.03·51-s + 4.80·53-s − 4.29·55-s + ⋯ |
| L(s) = 1 | − 1.81·3-s − 0.447·5-s + 1.47·7-s + 2.29·9-s + 1.29·11-s + 1.20·13-s + 0.812·15-s + 0.388·17-s + 0.275·19-s − 2.67·21-s + 1.74·23-s + 0.200·25-s − 2.35·27-s − 0.185·29-s + 0.430·31-s − 2.34·33-s − 0.657·35-s − 1.60·37-s − 2.18·39-s + 0.903·41-s − 0.549·43-s − 1.02·45-s − 1.25·47-s + 1.16·49-s − 0.705·51-s + 0.659·53-s − 0.578·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363774055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363774055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 9.78T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 + 7.55T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 + 0.348T + 97T^{2} \) |
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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
−8.911509985105241525149901627989, −8.225555723564019612679746299702, −7.13610196264072517862210565184, −6.71229722605667698273601352902, −5.73360341344603650582517516797, −5.10795396701565218452540737344, −4.41153896585290845959839962050, −3.58513692295709829709026224148, −1.51091596919083979915584527398, −0.976472980196388088383635668060,
0.976472980196388088383635668060, 1.51091596919083979915584527398, 3.58513692295709829709026224148, 4.41153896585290845959839962050, 5.10795396701565218452540737344, 5.73360341344603650582517516797, 6.71229722605667698273601352902, 7.13610196264072517862210565184, 8.225555723564019612679746299702, 8.911509985105241525149901627989
