L(s) = 1 | + 2.33·2-s − 3-s + 3.42·4-s − 0.900·5-s − 2.33·6-s − 7-s + 3.33·8-s + 9-s − 2.09·10-s − 2.90·11-s − 3.42·12-s − 2.33·14-s + 0.900·15-s + 0.900·16-s + 1.66·17-s + 2.33·18-s + 3.66·19-s − 3.08·20-s + 21-s − 6.75·22-s − 4.42·23-s − 3.33·24-s − 4.18·25-s − 27-s − 3.42·28-s + 5.33·29-s + 2.09·30-s + ⋯ |
L(s) = 1 | + 1.64·2-s − 0.577·3-s + 1.71·4-s − 0.402·5-s − 0.951·6-s − 0.377·7-s + 1.17·8-s + 0.333·9-s − 0.663·10-s − 0.874·11-s − 0.989·12-s − 0.622·14-s + 0.232·15-s + 0.225·16-s + 0.405·17-s + 0.549·18-s + 0.839·19-s − 0.690·20-s + 0.218·21-s − 1.44·22-s − 0.923·23-s − 0.679·24-s − 0.837·25-s − 0.192·27-s − 0.648·28-s + 0.989·29-s + 0.383·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 + 0.900T + 5T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 + 6.95T + 47T^{2} \) |
| 53 | \( 1 + 0.141T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 2.51T + 61T^{2} \) |
| 67 | \( 1 + 0.00982T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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
−7.75854101571893043557721005708, −7.30520047000748538644202683177, −6.35833340594431552586895716601, −5.76741710326397795559663014061, −5.15621284134338230131627632121, −4.43330818439786631827919132617, −3.58253861221764726630763102922, −2.95717333945694105747632817977, −1.79160473944103071544776913704, 0,
1.79160473944103071544776913704, 2.95717333945694105747632817977, 3.58253861221764726630763102922, 4.43330818439786631827919132617, 5.15621284134338230131627632121, 5.76741710326397795559663014061, 6.35833340594431552586895716601, 7.30520047000748538644202683177, 7.75854101571893043557721005708