| L(s) = 1 | + 3·3-s − 7-s + 6·9-s + 6·11-s − 13-s − 3·21-s + 23-s − 5·25-s + 9·27-s + 3·29-s − 3·31-s + 18·33-s + 8·37-s − 3·39-s + 9·41-s − 4·43-s + 13·47-s + 49-s − 4·53-s − 4·59-s − 2·61-s − 6·63-s + 4·67-s + 3·69-s − 5·71-s + 3·73-s − 15·75-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s + 1.80·11-s − 0.277·13-s − 0.654·21-s + 0.208·23-s − 25-s + 1.73·27-s + 0.557·29-s − 0.538·31-s + 3.13·33-s + 1.31·37-s − 0.480·39-s + 1.40·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.549·53-s − 0.520·59-s − 0.256·61-s − 0.755·63-s + 0.488·67-s + 0.361·69-s − 0.593·71-s + 0.351·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.804259393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.804259393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.49920353122393, −15.90961633036373, −15.21925333941386, −14.84792075350334, −14.22355276636846, −13.93563506669691, −13.33780460125921, −12.61939475934253, −12.16012556407347, −11.40988360698195, −10.65153781275297, −9.676327194822057, −9.508019596516539, −8.990703409057445, −8.359302629674740, −7.668656335628175, −7.138940903151077, −6.445733966118529, −5.740906032112891, −4.409787487135721, −4.074265248892411, −3.374373574003459, −2.644345457860027, −1.881515618768802, −0.9983685276624681,
0.9983685276624681, 1.881515618768802, 2.644345457860027, 3.374373574003459, 4.074265248892411, 4.409787487135721, 5.740906032112891, 6.445733966118529, 7.138940903151077, 7.668656335628175, 8.359302629674740, 8.990703409057445, 9.508019596516539, 9.676327194822057, 10.65153781275297, 11.40988360698195, 12.16012556407347, 12.61939475934253, 13.33780460125921, 13.93563506669691, 14.22355276636846, 14.84792075350334, 15.21925333941386, 15.90961633036373, 16.49920353122393
