| L(s) = 1 | + 2-s + 2.61·3-s + 4-s + 2.61·6-s + 1.61·7-s + 8-s + 3.85·9-s + 11-s + 2.61·12-s + 1.23·13-s + 1.61·14-s + 16-s + 0.763·17-s + 3.85·18-s + 4.23·21-s + 22-s + 0.381·23-s + 2.61·24-s + 1.23·26-s + 2.23·27-s + 1.61·28-s − 3.09·29-s + 2·31-s + 32-s + 2.61·33-s + 0.763·34-s + 3.85·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 1.06·6-s + 0.611·7-s + 0.353·8-s + 1.28·9-s + 0.301·11-s + 0.755·12-s + 0.342·13-s + 0.432·14-s + 0.250·16-s + 0.185·17-s + 0.908·18-s + 0.924·21-s + 0.213·22-s + 0.0796·23-s + 0.534·24-s + 0.242·26-s + 0.430·27-s + 0.305·28-s − 0.573·29-s + 0.359·31-s + 0.176·32-s + 0.455·33-s + 0.131·34-s + 0.642·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.489349563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.489349563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.381T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 0.909T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 0.0901T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 5.70T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 0.944T + 97T^{2} \) |
| 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
−8.575509505239067890168223835406, −8.190993133586674995743693269901, −7.37647521384077983434298589026, −6.66040837614641463550074002904, −5.61083243267716322283653389041, −4.72186208445271243408845837065, −3.85599478951911115318034105907, −3.24778173877597201328236647586, −2.28942256594814126408427138360, −1.46018589349766774281830868782,
1.46018589349766774281830868782, 2.28942256594814126408427138360, 3.24778173877597201328236647586, 3.85599478951911115318034105907, 4.72186208445271243408845837065, 5.61083243267716322283653389041, 6.66040837614641463550074002904, 7.37647521384077983434298589026, 8.190993133586674995743693269901, 8.575509505239067890168223835406
