| L(s) = 1 | − 2·4-s − 0.102·7-s − 7.09·13-s + 4·16-s + 5.89·19-s + 0.204·28-s + 6.07·31-s + 4.01·37-s + 3.08·43-s − 6.98·49-s + 14.1·52-s − 12.2·61-s − 8·64-s + 16.3·67-s − 16.2·73-s − 11.7·76-s + 15.2·79-s + 0.725·91-s − 16.5·97-s − 18.2·103-s + 10.2·109-s − 0.408·112-s + ⋯ |
| L(s) = 1 | − 4-s − 0.0386·7-s − 1.96·13-s + 16-s + 1.35·19-s + 0.0386·28-s + 1.09·31-s + 0.660·37-s + 0.469·43-s − 0.998·49-s + 1.96·52-s − 1.56·61-s − 64-s + 1.99·67-s − 1.90·73-s − 1.35·76-s + 1.71·79-s + 0.0760·91-s − 1.68·97-s − 1.79·103-s + 0.984·109-s − 0.0386·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 0.102T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 6.07T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
| 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
−7.76315480946105398550604951113, −7.28290772609383063782119907528, −6.31524421624910779920131128271, −5.37526420487367329933070828925, −4.89882396046079914095755712877, −4.25440336980744737392583851466, −3.23263150536723322050498846731, −2.49626856021769923801238615342, −1.13786011582789820758986333318, 0,
1.13786011582789820758986333318, 2.49626856021769923801238615342, 3.23263150536723322050498846731, 4.25440336980744737392583851466, 4.89882396046079914095755712877, 5.37526420487367329933070828925, 6.31524421624910779920131128271, 7.28290772609383063782119907528, 7.76315480946105398550604951113
