| L(s) = 1 | − 0.481·3-s + 0.806·7-s − 2.76·9-s + 3.67·11-s + 13-s + 1.35·17-s + 1.67·19-s − 0.387·21-s + 6.48·23-s + 2.77·27-s + 2.41·29-s + 5.28·31-s − 1.76·33-s − 3.76·37-s − 0.481·39-s − 8.31·41-s − 6.79·43-s + 3.19·47-s − 6.35·49-s − 0.649·51-s − 5.73·53-s − 0.806·57-s − 5.98·59-s − 1.76·61-s − 2.23·63-s + 9.89·67-s − 3.11·69-s + ⋯ |
| L(s) = 1 | − 0.277·3-s + 0.304·7-s − 0.922·9-s + 1.10·11-s + 0.277·13-s + 0.327·17-s + 0.384·19-s − 0.0846·21-s + 1.35·23-s + 0.534·27-s + 0.449·29-s + 0.949·31-s − 0.307·33-s − 0.619·37-s − 0.0770·39-s − 1.29·41-s − 1.03·43-s + 0.465·47-s − 0.907·49-s − 0.0909·51-s − 0.788·53-s − 0.106·57-s − 0.779·59-s − 0.226·61-s − 0.281·63-s + 1.20·67-s − 0.375·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.893404310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893404310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.481T + 3T^{2} \) |
| 7 | \( 1 - 0.806T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 + 6.79T + 43T^{2} \) |
| 47 | \( 1 - 3.19T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
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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
−8.374345871705364652640944262620, −7.47198922419309396174156627003, −6.58707741990930845938164563558, −6.21541188171850920208144175591, −5.16389712264865710097337939401, −4.76182917496978092018793814854, −3.55054828953838258955256767161, −3.02843796566918960390976846472, −1.75203963362488086955795386363, −0.78738519042896062409453744008,
0.78738519042896062409453744008, 1.75203963362488086955795386363, 3.02843796566918960390976846472, 3.55054828953838258955256767161, 4.76182917496978092018793814854, 5.16389712264865710097337939401, 6.21541188171850920208144175591, 6.58707741990930845938164563558, 7.47198922419309396174156627003, 8.374345871705364652640944262620
