L(s) = 1 | + 3.60·5-s − 7-s − 3.60·11-s + 6·13-s + 7.21·17-s − 19-s − 3.60·23-s + 7.99·25-s − 7.21·29-s + 9·31-s − 3.60·35-s − 37-s − 10.8·41-s + 8·43-s + 49-s − 12.9·55-s + 14.4·59-s + 21.6·65-s − 2·67-s − 3.60·71-s + 4·73-s + 3.60·77-s + 7.21·83-s + 25.9·85-s − 10.8·89-s − 6·91-s − 3.60·95-s + ⋯ |
L(s) = 1 | + 1.61·5-s − 0.377·7-s − 1.08·11-s + 1.66·13-s + 1.74·17-s − 0.229·19-s − 0.751·23-s + 1.59·25-s − 1.33·29-s + 1.61·31-s − 0.609·35-s − 0.164·37-s − 1.68·41-s + 1.21·43-s + 0.142·49-s − 1.75·55-s + 1.87·59-s + 2.68·65-s − 0.244·67-s − 0.427·71-s + 0.468·73-s + 0.410·77-s + 0.791·83-s + 2.82·85-s − 1.14·89-s − 0.628·91-s − 0.369·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.004286504\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.004286504\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.21T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.15865821279539791842837493626, −9.751917983836325808114755784613, −8.676786905449867073397728073977, −7.88404450465147372410417010892, −6.59973805497420624378589710218, −5.80049471794337427568904715192, −5.34601532650038842843207061648, −3.71486591510770636319113262187, −2.59629463337659630439550089133, −1.35148341909710539637175044299,
1.35148341909710539637175044299, 2.59629463337659630439550089133, 3.71486591510770636319113262187, 5.34601532650038842843207061648, 5.80049471794337427568904715192, 6.59973805497420624378589710218, 7.88404450465147372410417010892, 8.676786905449867073397728073977, 9.751917983836325808114755784613, 10.15865821279539791842837493626