| L(s) = 1 | − 2-s + 3-s + 4-s + 2.56·5-s − 6-s + 3.26·7-s − 8-s + 9-s − 2.56·10-s + 0.513·11-s + 12-s − 3.20·13-s − 3.26·14-s + 2.56·15-s + 16-s − 2.52·17-s − 18-s + 3.58·19-s + 2.56·20-s + 3.26·21-s − 0.513·22-s + 23-s − 24-s + 1.56·25-s + 3.20·26-s + 27-s + 3.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 1.23·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s + 0.154·11-s + 0.288·12-s − 0.889·13-s − 0.871·14-s + 0.661·15-s + 0.250·16-s − 0.611·17-s − 0.235·18-s + 0.821·19-s + 0.572·20-s + 0.711·21-s − 0.109·22-s + 0.208·23-s − 0.204·24-s + 0.312·25-s + 0.628·26-s + 0.192·27-s + 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.541595469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.541595469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 0.513T + 11T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 - 6.04T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 - 0.354T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 - 2.85T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.616951766616625472875110681169, −7.61123147933625643433077490108, −7.37715387821468825447172941392, −6.32363335781910187642696462101, −5.46598733147005851199298878677, −4.83884406080043666699980820402, −3.74777876793980576978771105976, −2.42266334594031928169210101362, −2.07737583959474543666899290444, −1.04303838184441728380202124826,
1.04303838184441728380202124826, 2.07737583959474543666899290444, 2.42266334594031928169210101362, 3.74777876793980576978771105976, 4.83884406080043666699980820402, 5.46598733147005851199298878677, 6.32363335781910187642696462101, 7.37715387821468825447172941392, 7.61123147933625643433077490108, 8.616951766616625472875110681169
