L(s) = 1 | + 2-s + 3-s + 4-s + 2.81·5-s + 6-s − 7-s + 8-s + 9-s + 2.81·10-s + 5.51·11-s + 12-s − 13-s − 14-s + 2.81·15-s + 16-s + 17-s + 18-s + 0.171·19-s + 2.81·20-s − 21-s + 5.51·22-s + 1.94·23-s + 24-s + 2.90·25-s − 26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.25·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.889·10-s + 1.66·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.725·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.0393·19-s + 0.628·20-s − 0.218·21-s + 1.17·22-s + 0.405·23-s + 0.204·24-s + 0.581·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.139211864\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.139211864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2.81T + 5T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 19 | \( 1 - 0.171T + 19T^{2} \) |
| 23 | \( 1 - 1.94T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 + 0.545T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 2.29T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−7.51130259617657337917629936615, −6.78736175089026054686264872777, −6.41040125100368587271621173627, −5.62854300687229214998662657304, −5.05568917400132990619707181754, −3.98506473357320464247805606776, −3.59432623228594417130631700745, −2.59368523851522535956565555060, −1.92857615532574497930583453537, −1.12664025041173642566063302503,
1.12664025041173642566063302503, 1.92857615532574497930583453537, 2.59368523851522535956565555060, 3.59432623228594417130631700745, 3.98506473357320464247805606776, 5.05568917400132990619707181754, 5.62854300687229214998662657304, 6.41040125100368587271621173627, 6.78736175089026054686264872777, 7.51130259617657337917629936615