| L(s) = 1 | − 2.74·2-s + 1.79·3-s + 5.54·4-s − 1.53·5-s − 4.93·6-s − 2.66·7-s − 9.75·8-s + 0.228·9-s + 4.22·10-s + 6.00·11-s + 9.97·12-s + 2.63·13-s + 7.31·14-s − 2.76·15-s + 15.6·16-s − 2.14·17-s − 0.628·18-s − 5.68·19-s − 8.53·20-s − 4.78·21-s − 16.5·22-s + 5.77·23-s − 17.5·24-s − 2.63·25-s − 7.24·26-s − 4.97·27-s − 14.7·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.03·3-s + 2.77·4-s − 0.687·5-s − 2.01·6-s − 1.00·7-s − 3.44·8-s + 0.0762·9-s + 1.33·10-s + 1.81·11-s + 2.87·12-s + 0.731·13-s + 1.95·14-s − 0.713·15-s + 3.92·16-s − 0.519·17-s − 0.148·18-s − 1.30·19-s − 1.90·20-s − 1.04·21-s − 3.51·22-s + 1.20·23-s − 3.57·24-s − 0.527·25-s − 1.42·26-s − 0.958·27-s − 2.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7733066390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7733066390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.66T + 7T^{2} \) |
| 11 | \( 1 - 6.00T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 0.852T + 37T^{2} \) |
| 41 | \( 1 + 7.11T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 0.221T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.650574731989685156340662480185, −7.70548426980374902088434887927, −7.04972294360673188108128801846, −6.52303888372061865661017944753, −5.88285557545073284673260065915, −3.86461787525823400503109101976, −3.59588068499148483771572631908, −2.55567581225218638026585417830, −1.75361729251070713300759993340, −0.59945465871460042640236685829,
0.59945465871460042640236685829, 1.75361729251070713300759993340, 2.55567581225218638026585417830, 3.59588068499148483771572631908, 3.86461787525823400503109101976, 5.88285557545073284673260065915, 6.52303888372061865661017944753, 7.04972294360673188108128801846, 7.70548426980374902088434887927, 8.650574731989685156340662480185
