| L(s) = 1 | + 2.82·5-s + 6·11-s + 5.65·13-s + 1.41·17-s − 4.24·19-s + 4·23-s + 3.00·25-s + 6·29-s + 2.82·31-s + 2·37-s − 1.41·41-s − 10·43-s + 2.82·47-s + 2·53-s + 16.9·55-s − 1.41·59-s + 8.48·61-s + 16.0·65-s − 4·67-s − 12·71-s + 9.89·73-s + 4·79-s − 1.41·83-s + 4.00·85-s − 4.24·89-s − 12·95-s − 12.7·97-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 1.80·11-s + 1.56·13-s + 0.342·17-s − 0.973·19-s + 0.834·23-s + 0.600·25-s + 1.11·29-s + 0.508·31-s + 0.328·37-s − 0.220·41-s − 1.52·43-s + 0.412·47-s + 0.274·53-s + 2.28·55-s − 0.184·59-s + 1.08·61-s + 1.98·65-s − 0.488·67-s − 1.42·71-s + 1.15·73-s + 0.450·79-s − 0.155·83-s + 0.433·85-s − 0.449·89-s − 1.23·95-s − 1.29·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550358959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550358959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.186606903107144253527389982554, −6.86449818490525449453570564035, −6.48807244167066257063252337390, −6.04388864948741487810013134854, −5.22714853778186210804765791226, −4.27212020799128708050598395436, −3.62953030193807424433779539844, −2.66961043669717636699923991565, −1.57661125791959474201251567119, −1.11981705315670534212033177055,
1.11981705315670534212033177055, 1.57661125791959474201251567119, 2.66961043669717636699923991565, 3.62953030193807424433779539844, 4.27212020799128708050598395436, 5.22714853778186210804765791226, 6.04388864948741487810013134854, 6.48807244167066257063252337390, 6.86449818490525449453570564035, 8.186606903107144253527389982554
