| L(s) = 1 | + 1.39·2-s − 0.528·3-s − 0.0429·4-s − 5-s − 0.738·6-s − 3.55·7-s − 2.85·8-s − 2.72·9-s − 1.39·10-s + 11-s + 0.0227·12-s − 3.96·13-s − 4.97·14-s + 0.528·15-s − 3.91·16-s + 3.39·17-s − 3.80·18-s − 2.71·19-s + 0.0429·20-s + 1.87·21-s + 1.39·22-s + 3.30·23-s + 1.50·24-s + 25-s − 5.54·26-s + 3.02·27-s + 0.152·28-s + ⋯ |
| L(s) = 1 | + 0.989·2-s − 0.304·3-s − 0.0214·4-s − 0.447·5-s − 0.301·6-s − 1.34·7-s − 1.01·8-s − 0.907·9-s − 0.442·10-s + 0.301·11-s + 0.00655·12-s − 1.09·13-s − 1.33·14-s + 0.136·15-s − 0.978·16-s + 0.822·17-s − 0.897·18-s − 0.622·19-s + 0.00961·20-s + 0.410·21-s + 0.298·22-s + 0.689·23-s + 0.308·24-s + 0.200·25-s − 1.08·26-s + 0.581·27-s + 0.0289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7713282578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7713282578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 0.528T + 3T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 + 9.88T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 + 0.353T + 61T^{2} \) |
| 67 | \( 1 - 9.25T + 67T^{2} \) |
| 71 | \( 1 + 0.985T + 71T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 0.167T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
| 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
−8.538138664582100763052387226889, −7.48650674446451698445222018515, −6.79499600786398303889220412315, −6.01244131549180989861786229978, −5.47519568351122367046736104576, −4.73223854499021731862900640979, −3.73107568612510862711554618199, −3.26818501387540833559275875577, −2.42259312740811218122396241237, −0.40990179274724169868546520436,
0.40990179274724169868546520436, 2.42259312740811218122396241237, 3.26818501387540833559275875577, 3.73107568612510862711554618199, 4.73223854499021731862900640979, 5.47519568351122367046736104576, 6.01244131549180989861786229978, 6.79499600786398303889220412315, 7.48650674446451698445222018515, 8.538138664582100763052387226889
