| L(s) = 1 | + 0.170·2-s + 2.94·3-s − 1.97·4-s − 1.29·5-s + 0.503·6-s − 3.65·7-s − 0.678·8-s + 5.69·9-s − 0.221·10-s + 11-s − 5.81·12-s − 0.624·14-s − 3.82·15-s + 3.82·16-s + 3.48·17-s + 0.973·18-s − 2.45·19-s + 2.55·20-s − 10.7·21-s + 0.170·22-s − 4.00·23-s − 2.00·24-s − 3.31·25-s + 7.94·27-s + 7.19·28-s − 8.42·29-s − 0.654·30-s + ⋯ |
| L(s) = 1 | + 0.120·2-s + 1.70·3-s − 0.985·4-s − 0.580·5-s + 0.205·6-s − 1.38·7-s − 0.239·8-s + 1.89·9-s − 0.0701·10-s + 0.301·11-s − 1.67·12-s − 0.166·14-s − 0.988·15-s + 0.956·16-s + 0.844·17-s + 0.229·18-s − 0.564·19-s + 0.572·20-s − 2.35·21-s + 0.0364·22-s − 0.836·23-s − 0.408·24-s − 0.662·25-s + 1.52·27-s + 1.36·28-s − 1.56·29-s − 0.119·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.170T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 + 9.21T + 31T^{2} \) |
| 37 | \( 1 - 0.0815T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 0.725T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 - 6.01T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 5.30T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 - 0.0792T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
| 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
−9.033397117213222632207219745261, −7.952264922535482457104290902295, −7.75368086319389312376752151953, −6.59167878879219982126839661542, −5.57770853029370933152559310520, −4.24727566929236068385461438277, −3.59031670140925348418104174323, −3.27629967522772087466842056711, −1.87736043397180175941583188941, 0,
1.87736043397180175941583188941, 3.27629967522772087466842056711, 3.59031670140925348418104174323, 4.24727566929236068385461438277, 5.57770853029370933152559310520, 6.59167878879219982126839661542, 7.75368086319389312376752151953, 7.952264922535482457104290902295, 9.033397117213222632207219745261
