| L(s) = 1 | + 1.97·2-s + 0.160·3-s + 1.91·4-s − 3.06·5-s + 0.317·6-s + 2.43·7-s − 0.160·8-s − 2.97·9-s − 6.05·10-s − 2.02·11-s + 0.307·12-s + 2.03·13-s + 4.81·14-s − 0.490·15-s − 4.15·16-s + 5.09·17-s − 5.88·18-s − 1.84·19-s − 5.87·20-s + 0.390·21-s − 4.00·22-s − 8.84·23-s − 0.0257·24-s + 4.36·25-s + 4.02·26-s − 0.958·27-s + 4.66·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.0925·3-s + 0.959·4-s − 1.36·5-s + 0.129·6-s + 0.919·7-s − 0.0567·8-s − 0.991·9-s − 1.91·10-s − 0.610·11-s + 0.0888·12-s + 0.563·13-s + 1.28·14-s − 0.126·15-s − 1.03·16-s + 1.23·17-s − 1.38·18-s − 0.423·19-s − 1.31·20-s + 0.0851·21-s − 0.854·22-s − 1.84·23-s − 0.00525·24-s + 0.873·25-s + 0.788·26-s − 0.184·27-s + 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 - 0.160T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 - 0.708T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 - 0.861T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 6.37T + 83T^{2} \) |
| 89 | \( 1 - 0.792T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.482507243880880299611008074426, −7.983649119499377371350864889801, −7.39428212015055371944382438794, −6.00234231510666615891135327738, −5.56480559162373784871234678339, −4.62744297435923650871712143293, −3.81915334452722141870811074806, −3.30739119182396557210180334320, −2.06548256552705972178437134905, 0,
2.06548256552705972178437134905, 3.30739119182396557210180334320, 3.81915334452722141870811074806, 4.62744297435923650871712143293, 5.56480559162373784871234678339, 6.00234231510666615891135327738, 7.39428212015055371944382438794, 7.983649119499377371350864889801, 8.482507243880880299611008074426
