L(s) = 1 | − 0.428·2-s − 2.33·3-s − 1.81·4-s + 0.998·6-s + 3.85·7-s + 1.63·8-s + 2.43·9-s − 0.897·11-s + 4.23·12-s + 4.05·13-s − 1.65·14-s + 2.93·16-s + 3.01·17-s − 1.04·18-s − 5.78·19-s − 8.99·21-s + 0.384·22-s + 7.46·23-s − 3.81·24-s − 1.73·26-s + 1.31·27-s − 7.00·28-s + 9.45·29-s + 8.89·31-s − 4.52·32-s + 2.09·33-s − 1.29·34-s + ⋯ |
L(s) = 1 | − 0.302·2-s − 1.34·3-s − 0.908·4-s + 0.407·6-s + 1.45·7-s + 0.577·8-s + 0.811·9-s − 0.270·11-s + 1.22·12-s + 1.12·13-s − 0.441·14-s + 0.733·16-s + 0.732·17-s − 0.245·18-s − 1.32·19-s − 1.96·21-s + 0.0819·22-s + 1.55·23-s − 0.777·24-s − 0.340·26-s + 0.253·27-s − 1.32·28-s + 1.75·29-s + 1.59·31-s − 0.799·32-s + 0.364·33-s − 0.221·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.231883141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.231883141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.428T + 2T^{2} \) |
| 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 0.897T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 19 | \( 1 + 5.78T + 19T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 - 9.45T + 29T^{2} \) |
| 31 | \( 1 - 8.89T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.57T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 6.63T + 97T^{2} \) |
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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
−8.279388425300818322647631111535, −7.50310229651440589207165804611, −6.41351044320992870502346343175, −5.95796972564006637240263065835, −4.96757734536491427877108169952, −4.76925637777911051807827653940, −4.04352935201219988662632506653, −2.70817880037872178440536602405, −1.19650731321265511077735403147, −0.846438957358517436873463481500,
0.846438957358517436873463481500, 1.19650731321265511077735403147, 2.70817880037872178440536602405, 4.04352935201219988662632506653, 4.76925637777911051807827653940, 4.96757734536491427877108169952, 5.95796972564006637240263065835, 6.41351044320992870502346343175, 7.50310229651440589207165804611, 8.279388425300818322647631111535