| L(s) = 1 | + 2.54·3-s + 3.96·5-s + 3.49·9-s + 4.69·11-s + 0.0691·13-s + 10.0·15-s − 6.19·17-s − 3.02·19-s + 9.15·23-s + 10.7·25-s + 1.25·27-s − 6.93·29-s − 2.39·31-s + 11.9·33-s − 2.02·37-s + 0.176·39-s + 7.07·41-s − 0.137·43-s + 13.8·45-s + 5.91·47-s − 15.7·51-s − 4.75·53-s + 18.5·55-s − 7.70·57-s − 3.73·59-s − 0.224·61-s + 0.274·65-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.77·5-s + 1.16·9-s + 1.41·11-s + 0.0191·13-s + 2.60·15-s − 1.50·17-s − 0.693·19-s + 1.90·23-s + 2.14·25-s + 0.242·27-s − 1.28·29-s − 0.429·31-s + 2.08·33-s − 0.332·37-s + 0.0282·39-s + 1.10·41-s − 0.0209·43-s + 2.06·45-s + 0.863·47-s − 2.21·51-s − 0.652·53-s + 2.50·55-s − 1.02·57-s − 0.486·59-s − 0.0287·61-s + 0.0340·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.272701783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.272701783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 - 3.96T + 5T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 0.0691T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 9.15T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 + 2.02T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 0.137T + 43T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 + 0.224T + 61T^{2} \) |
| 67 | \( 1 - 5.82T + 67T^{2} \) |
| 71 | \( 1 - 6.85T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 - 6.03T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.547973974109146834688711194630, −7.35724613769466618458113396864, −6.74349447544498156457298033516, −6.19151387891157583661964888113, −5.26987211143743110797841939625, −4.35086996253325776447648935444, −3.55683598800896416696794794695, −2.58276150252140411335547245854, −2.06080189853647426734769217598, −1.29866538247718943765782887436,
1.29866538247718943765782887436, 2.06080189853647426734769217598, 2.58276150252140411335547245854, 3.55683598800896416696794794695, 4.35086996253325776447648935444, 5.26987211143743110797841939625, 6.19151387891157583661964888113, 6.74349447544498156457298033516, 7.35724613769466618458113396864, 8.547973974109146834688711194630
