| L(s) = 1 | − 1.36·3-s + 3.28·7-s − 1.13·9-s + 3.49·11-s − 3.41·13-s + 7.46·17-s + 3.49·19-s − 4.48·21-s − 23-s + 5.64·27-s − 3.46·29-s + 2.01·31-s − 4.77·33-s − 0.511·37-s + 4.66·39-s − 7.07·41-s + 2.76·43-s + 0.889·47-s + 3.76·49-s − 10.1·51-s + 14.2·53-s − 4.77·57-s − 4.71·59-s + 13.4·61-s − 3.71·63-s + 2.30·67-s + 1.36·69-s + ⋯ |
| L(s) = 1 | − 0.788·3-s + 1.24·7-s − 0.377·9-s + 1.05·11-s − 0.946·13-s + 1.80·17-s + 0.802·19-s − 0.978·21-s − 0.208·23-s + 1.08·27-s − 0.643·29-s + 0.361·31-s − 0.831·33-s − 0.0840·37-s + 0.746·39-s − 1.10·41-s + 0.421·43-s + 0.129·47-s + 0.537·49-s − 1.42·51-s + 1.95·53-s − 0.632·57-s − 0.613·59-s + 1.72·61-s − 0.468·63-s + 0.281·67-s + 0.164·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773350150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773350150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 0.511T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 0.889T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 - 0.801T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.283463863433849833058881172220, −7.50681828132356218298230114219, −6.98156000884469247769898666940, −5.87154812493552166486977784067, −5.43483164126271391838384377944, −4.80853747565466031015092006648, −3.88883917580553656383597929590, −2.90910733755853297359107939447, −1.69355216813757696917366459326, −0.821528322779340646549064741760,
0.821528322779340646549064741760, 1.69355216813757696917366459326, 2.90910733755853297359107939447, 3.88883917580553656383597929590, 4.80853747565466031015092006648, 5.43483164126271391838384377944, 5.87154812493552166486977784067, 6.98156000884469247769898666940, 7.50681828132356218298230114219, 8.283463863433849833058881172220
