| L(s) = 1 | + 3-s + 4.13·7-s + 9-s − 1.23·11-s + 5.04·13-s − 4.23·17-s + 3.14·19-s + 4.13·21-s − 6.74·23-s + 27-s − 6.24·29-s − 31-s − 1.23·33-s + 8.70·37-s + 5.04·39-s + 8.60·41-s − 10.8·43-s + 2.82·47-s + 10.1·49-s − 4.23·51-s + 4.85·53-s + 3.14·57-s + 7.88·59-s + 9.08·61-s + 4.13·63-s + 15.5·67-s − 6.74·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.56·7-s + 0.333·9-s − 0.370·11-s + 1.39·13-s − 1.02·17-s + 0.722·19-s + 0.902·21-s − 1.40·23-s + 0.192·27-s − 1.16·29-s − 0.179·31-s − 0.214·33-s + 1.43·37-s + 0.808·39-s + 1.34·41-s − 1.64·43-s + 0.412·47-s + 1.44·49-s − 0.593·51-s + 0.666·53-s + 0.417·57-s + 1.02·59-s + 1.16·61-s + 0.520·63-s + 1.89·67-s − 0.812·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.452022440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.452022440\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 + 3.49T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.79522930286542110682509341488, −7.28744150541337888840340961634, −6.28717826235914292569551946126, −5.63606888783976101232830493334, −4.89127250738304340890624180434, −4.10193929268537845175051784601, −3.63217824895331283094594820440, −2.38155019072095182969061301446, −1.87425999405026257645961406433, −0.904355668250851425416019684834,
0.904355668250851425416019684834, 1.87425999405026257645961406433, 2.38155019072095182969061301446, 3.63217824895331283094594820440, 4.10193929268537845175051784601, 4.89127250738304340890624180434, 5.63606888783976101232830493334, 6.28717826235914292569551946126, 7.28744150541337888840340961634, 7.79522930286542110682509341488
