| L(s) = 1 | − 1.37·2-s − 1.23·3-s − 0.115·4-s + 5-s + 1.69·6-s + 2.43·7-s + 2.90·8-s − 1.47·9-s − 1.37·10-s − 11-s + 0.142·12-s − 4.84·13-s − 3.33·14-s − 1.23·15-s − 3.75·16-s − 1.04·17-s + 2.01·18-s + 19-s − 0.115·20-s − 3.00·21-s + 1.37·22-s + 0.377·23-s − 3.59·24-s + 25-s + 6.65·26-s + 5.52·27-s − 0.280·28-s + ⋯ |
| L(s) = 1 | − 0.970·2-s − 0.713·3-s − 0.0577·4-s + 0.447·5-s + 0.692·6-s + 0.919·7-s + 1.02·8-s − 0.490·9-s − 0.434·10-s − 0.301·11-s + 0.0412·12-s − 1.34·13-s − 0.892·14-s − 0.319·15-s − 0.938·16-s − 0.252·17-s + 0.476·18-s + 0.229·19-s − 0.0258·20-s − 0.656·21-s + 0.292·22-s + 0.0786·23-s − 0.732·24-s + 0.200·25-s + 1.30·26-s + 1.06·27-s − 0.0530·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 23 | \( 1 - 0.377T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 0.790T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 + 0.259T + 71T^{2} \) |
| 73 | \( 1 + 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−9.676762083652218684649954422926, −8.604562646497179037503950226613, −8.059568746557333462231091748069, −7.16376721065267333255689781297, −6.14578308566752869084308129119, −4.90727779567080300743424068860, −4.77236175693271245142212910116, −2.75598085987582934371435161871, −1.46912671416111265120393175103, 0,
1.46912671416111265120393175103, 2.75598085987582934371435161871, 4.77236175693271245142212910116, 4.90727779567080300743424068860, 6.14578308566752869084308129119, 7.16376721065267333255689781297, 8.059568746557333462231091748069, 8.604562646497179037503950226613, 9.676762083652218684649954422926
