| L(s) = 1 | − 3.23·3-s + 2.23·5-s + 3.23·7-s + 7.47·9-s + 1.76·13-s − 7.23·15-s + 17-s − 5.70·19-s − 10.4·21-s + 0.763·23-s − 14.4·27-s + 1.76·29-s + 4.76·31-s + 7.23·35-s − 0.236·37-s − 5.70·39-s + 7.47·41-s + 10.4·43-s + 16.7·45-s + 5.70·47-s + 3.47·49-s − 3.23·51-s − 13.1·53-s + 18.4·57-s − 5.52·59-s + 14.9·61-s + 24.1·63-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 0.999·5-s + 1.22·7-s + 2.49·9-s + 0.489·13-s − 1.86·15-s + 0.242·17-s − 1.30·19-s − 2.28·21-s + 0.159·23-s − 2.78·27-s + 0.327·29-s + 0.855·31-s + 1.22·35-s − 0.0388·37-s − 0.914·39-s + 1.16·41-s + 1.59·43-s + 2.49·45-s + 0.832·47-s + 0.496·49-s − 0.453·51-s − 1.81·53-s + 2.44·57-s − 0.719·59-s + 1.91·61-s + 3.04·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224627343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224627343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 + 0.236T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−10.31846294866429154463528586243, −9.446177358737667417530547865169, −8.288445990407859052952823952227, −7.26108951946828757838567593700, −6.19674225091174843181567445059, −5.86114259675181829736928730372, −4.89308847686214655907872192021, −4.25464656987388789999573726974, −2.07899325679545239886163437291, −1.02521012388011631360463105375,
1.02521012388011631360463105375, 2.07899325679545239886163437291, 4.25464656987388789999573726974, 4.89308847686214655907872192021, 5.86114259675181829736928730372, 6.19674225091174843181567445059, 7.26108951946828757838567593700, 8.288445990407859052952823952227, 9.446177358737667417530547865169, 10.31846294866429154463528586243
