| L(s) = 1 | − 3.01·3-s + 4.48·7-s + 6.10·9-s + 3.39·11-s − 6.49·13-s − 3.15·17-s − 1.11·19-s − 13.5·21-s + 4.98·23-s − 9.35·27-s + 0.354·29-s + 5.42·31-s − 10.2·33-s + 2.87·37-s + 19.6·39-s − 0.211·41-s − 0.847·43-s − 2.09·47-s + 13.0·49-s + 9.50·51-s + 9.06·53-s + 3.35·57-s + 3.46·59-s + 3.78·61-s + 27.3·63-s + 7.73·67-s − 15.0·69-s + ⋯ |
| L(s) = 1 | − 1.74·3-s + 1.69·7-s + 2.03·9-s + 1.02·11-s − 1.80·13-s − 0.764·17-s − 0.254·19-s − 2.95·21-s + 1.04·23-s − 1.80·27-s + 0.0659·29-s + 0.974·31-s − 1.78·33-s + 0.471·37-s + 3.13·39-s − 0.0330·41-s − 0.129·43-s − 0.304·47-s + 1.87·49-s + 1.33·51-s + 1.24·53-s + 0.443·57-s + 0.451·59-s + 0.484·61-s + 3.44·63-s + 0.944·67-s − 1.81·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037498710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037498710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.01T + 3T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 0.354T + 29T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 + 0.211T + 41T^{2} \) |
| 43 | \( 1 + 0.847T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 - 9.06T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 0.0503T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−10.22967107112080789203615807022, −9.307109874779712146883368797713, −8.194045146264873147356519406427, −7.16769043865944030227925991733, −6.63483316616915228765245408933, −5.46021367744418967326168074788, −4.81687255541543521495691303559, −4.31096963388915290782774110453, −2.15096214265904922896510184099, −0.900366202041063146447159861662,
0.900366202041063146447159861662, 2.15096214265904922896510184099, 4.31096963388915290782774110453, 4.81687255541543521495691303559, 5.46021367744418967326168074788, 6.63483316616915228765245408933, 7.16769043865944030227925991733, 8.194045146264873147356519406427, 9.307109874779712146883368797713, 10.22967107112080789203615807022
