| L(s) = 1 | − 3-s − 7-s + 9-s + 6.31·11-s + 6.96·13-s + 6.57·17-s − 3.73·19-s + 21-s + 5.73·23-s − 27-s − 2·29-s − 1.03·31-s − 6.31·33-s + 10.7·37-s − 6.96·39-s − 6.96·41-s − 5.92·43-s + 49-s − 6.57·51-s + 1.03·53-s + 3.73·57-s − 3.22·59-s − 13.8·61-s − 63-s − 4.77·67-s − 5.73·69-s + 8.23·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 0.333·9-s + 1.90·11-s + 1.93·13-s + 1.59·17-s − 0.857·19-s + 0.218·21-s + 1.19·23-s − 0.192·27-s − 0.371·29-s − 0.186·31-s − 1.09·33-s + 1.75·37-s − 1.11·39-s − 1.08·41-s − 0.903·43-s + 0.142·49-s − 0.920·51-s + 0.142·53-s + 0.495·57-s − 0.419·59-s − 1.77·61-s − 0.125·63-s − 0.583·67-s − 0.690·69-s + 0.977·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.078086791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078086791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 4.77T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 - 2.18T + 89T^{2} \) |
| 97 | \( 1 - 3.73T + 97T^{2} \) |
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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
−8.527094314992415994016958458334, −7.62236760803150753034998451203, −6.64346641880773483296526408942, −6.28729919496132713418942660297, −5.67076961193174245177174276128, −4.58028315804638050865538773773, −3.74395382566675955436993015259, −3.26199858446333215858434490670, −1.56219300253496594831719765406, −0.963907712723982093926808996824,
0.963907712723982093926808996824, 1.56219300253496594831719765406, 3.26199858446333215858434490670, 3.74395382566675955436993015259, 4.58028315804638050865538773773, 5.67076961193174245177174276128, 6.28729919496132713418942660297, 6.64346641880773483296526408942, 7.62236760803150753034998451203, 8.527094314992415994016958458334
