L(s) = 1 | + 1.78·2-s − 3-s + 1.19·4-s − 1.65·5-s − 1.78·6-s − 7-s − 1.43·8-s + 9-s − 2.95·10-s − 5.05·11-s − 1.19·12-s − 4.10·13-s − 1.78·14-s + 1.65·15-s − 4.96·16-s − 1.03·17-s + 1.78·18-s − 0.125·19-s − 1.97·20-s + 21-s − 9.03·22-s − 4.97·23-s + 1.43·24-s − 2.27·25-s − 7.33·26-s − 27-s − 1.19·28-s + ⋯ |
L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.598·4-s − 0.738·5-s − 0.729·6-s − 0.377·7-s − 0.507·8-s + 0.333·9-s − 0.933·10-s − 1.52·11-s − 0.345·12-s − 1.13·13-s − 0.477·14-s + 0.426·15-s − 1.24·16-s − 0.251·17-s + 0.421·18-s − 0.0288·19-s − 0.441·20-s + 0.218·21-s − 1.92·22-s − 1.03·23-s + 0.293·24-s − 0.454·25-s − 1.43·26-s − 0.192·27-s − 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6002082884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6002082884\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 0.125T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 + 1.22T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 5.40T + 97T^{2} \) |
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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
−7.82695234848649630271258565620, −6.94732861761396013407522046931, −6.22449159148914528338870484787, −5.61353654899582655674106504074, −4.89986862317983795987437461954, −4.47983503983328014004507279075, −3.68744185882523675521367777333, −2.85898798031274507776582926657, −2.18844387693079548150403460295, −0.29878851265361664226308007846,
0.29878851265361664226308007846, 2.18844387693079548150403460295, 2.85898798031274507776582926657, 3.68744185882523675521367777333, 4.47983503983328014004507279075, 4.89986862317983795987437461954, 5.61353654899582655674106504074, 6.22449159148914528338870484787, 6.94732861761396013407522046931, 7.82695234848649630271258565620