| L(s) = 1 | − 2-s − 3-s + 4-s − 3.67·5-s + 6-s + 2.91·7-s − 8-s + 9-s + 3.67·10-s − 3.67·11-s − 12-s + 5.67·13-s − 2.91·14-s + 3.67·15-s + 16-s + 4·17-s − 18-s + 1.08·19-s − 3.67·20-s − 2.91·21-s + 3.67·22-s + 23-s + 24-s + 8.49·25-s − 5.67·26-s − 27-s + 2.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.64·5-s + 0.408·6-s + 1.10·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s − 1.10·11-s − 0.288·12-s + 1.57·13-s − 0.778·14-s + 0.948·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 0.249·19-s − 0.821·20-s − 0.635·21-s + 0.783·22-s + 0.208·23-s + 0.204·24-s + 1.69·25-s − 1.11·26-s − 0.192·27-s + 0.550·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8329417764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8329417764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 31 | \( 1 - 5.67T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 1.49T + 41T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 - 3.34T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 5.67T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 + 3.82T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−8.225721920056017391746272525302, −7.88679272366685201293298852401, −7.30001999135398930906216503357, −6.37026963216368250813404314290, −5.40833364209849351770907831376, −4.75425119792789616068195978081, −3.80281050045057767167130319274, −3.05937571786817834767683476403, −1.55798080964688856686629131892, −0.63869863352458806607521440643,
0.63869863352458806607521440643, 1.55798080964688856686629131892, 3.05937571786817834767683476403, 3.80281050045057767167130319274, 4.75425119792789616068195978081, 5.40833364209849351770907831376, 6.37026963216368250813404314290, 7.30001999135398930906216503357, 7.88679272366685201293298852401, 8.225721920056017391746272525302
