| L(s) = 1 | + 1.57·2-s − 2.87·3-s + 0.494·4-s − 4.53·6-s − 1.42·7-s − 2.37·8-s + 5.24·9-s − 0.0740·11-s − 1.42·12-s − 5.70·13-s − 2.24·14-s − 4.74·16-s + 8.29·18-s − 4.90·19-s + 4.08·21-s − 0.116·22-s − 3.88·23-s + 6.82·24-s − 9.00·26-s − 6.46·27-s − 0.702·28-s − 5.91·29-s − 0.388·31-s − 2.73·32-s + 0.212·33-s + 2.59·36-s − 9.91·37-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 1.65·3-s + 0.247·4-s − 1.85·6-s − 0.536·7-s − 0.840·8-s + 1.74·9-s − 0.0223·11-s − 0.410·12-s − 1.58·13-s − 0.599·14-s − 1.18·16-s + 1.95·18-s − 1.12·19-s + 0.890·21-s − 0.0249·22-s − 0.809·23-s + 1.39·24-s − 1.76·26-s − 1.24·27-s − 0.132·28-s − 1.09·29-s − 0.0697·31-s − 0.484·32-s + 0.0370·33-s + 0.432·36-s − 1.62·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2155899813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2155899813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 + 0.0740T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 0.388T + 31T^{2} \) |
| 37 | \( 1 + 9.91T + 37T^{2} \) |
| 41 | \( 1 - 6.61T + 41T^{2} \) |
| 43 | \( 1 - 6.94T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 + 0.0216T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 + 3.84T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 1.05T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 1.99T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47502191902226990355002726052, −6.93565405947524067888264592973, −6.18329957080066967936330491918, −5.77471836637819899294204594838, −5.11302811207022178907863060787, −4.50866565558689902509461754978, −3.95701940656231920615422188950, −2.89678617273194153296097916513, −1.88918383316780615893028055169, −0.20311218518310464062935841382,
0.20311218518310464062935841382, 1.88918383316780615893028055169, 2.89678617273194153296097916513, 3.95701940656231920615422188950, 4.50866565558689902509461754978, 5.11302811207022178907863060787, 5.77471836637819899294204594838, 6.18329957080066967936330491918, 6.93565405947524067888264592973, 7.47502191902226990355002726052
