| L(s) = 1 | + 1.39·2-s + 2.70·3-s − 0.0639·4-s − 0.456·5-s + 3.76·6-s + 1.81·7-s − 2.87·8-s + 4.33·9-s − 0.635·10-s + 1.42·11-s − 0.173·12-s − 13-s + 2.52·14-s − 1.23·15-s − 3.86·16-s − 0.961·17-s + 6.03·18-s + 0.499·19-s + 0.0292·20-s + 4.90·21-s + 1.98·22-s − 3.09·23-s − 7.78·24-s − 4.79·25-s − 1.39·26-s + 3.62·27-s − 0.115·28-s + ⋯ |
| L(s) = 1 | + 0.983·2-s + 1.56·3-s − 0.0319·4-s − 0.204·5-s + 1.53·6-s + 0.684·7-s − 1.01·8-s + 1.44·9-s − 0.201·10-s + 0.431·11-s − 0.0500·12-s − 0.277·13-s + 0.673·14-s − 0.319·15-s − 0.966·16-s − 0.233·17-s + 1.42·18-s + 0.114·19-s + 0.00653·20-s + 1.07·21-s + 0.424·22-s − 0.644·23-s − 1.58·24-s − 0.958·25-s − 0.272·26-s + 0.698·27-s − 0.0219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.115781155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.115781155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 + 0.456T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 17 | \( 1 + 0.961T + 17T^{2} \) |
| 19 | \( 1 - 0.499T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 + 4.01T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 7.10T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 + 0.812T + 83T^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−11.63970618787185733555117381783, −10.17920066150743349523398507775, −9.237031438456811547360190610808, −8.487631395910216202127464349560, −7.75802650472162642865578857830, −6.50739093008637731582180958026, −5.09717185831902291759656185891, −4.12668987108114259316458270442, −3.32394840403678948437393284703, −2.08649918029016325417391065340,
2.08649918029016325417391065340, 3.32394840403678948437393284703, 4.12668987108114259316458270442, 5.09717185831902291759656185891, 6.50739093008637731582180958026, 7.75802650472162642865578857830, 8.487631395910216202127464349560, 9.237031438456811547360190610808, 10.17920066150743349523398507775, 11.63970618787185733555117381783
