| L(s) = 1 | − 2-s − 3-s + 4-s − 4.37·5-s + 6-s − 0.773·7-s − 8-s + 9-s + 4.37·10-s + 4.42·11-s − 12-s + 0.648·13-s + 0.773·14-s + 4.37·15-s + 16-s + 6.40·17-s − 18-s − 19-s − 4.37·20-s + 0.773·21-s − 4.42·22-s + 0.287·23-s + 24-s + 14.1·25-s − 0.648·26-s − 27-s − 0.773·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.95·5-s + 0.408·6-s − 0.292·7-s − 0.353·8-s + 0.333·9-s + 1.38·10-s + 1.33·11-s − 0.288·12-s + 0.179·13-s + 0.206·14-s + 1.12·15-s + 0.250·16-s + 1.55·17-s − 0.235·18-s − 0.229·19-s − 0.978·20-s + 0.168·21-s − 0.942·22-s + 0.0599·23-s + 0.204·24-s + 2.82·25-s − 0.127·26-s − 0.192·27-s − 0.146·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7230734711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7230734711\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 + 0.773T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 0.648T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 23 | \( 1 - 0.287T + 23T^{2} \) |
| 29 | \( 1 - 0.347T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 8.14T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 + 1.42T + 71T^{2} \) |
| 73 | \( 1 + 8.80T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 9.18T + 97T^{2} \) |
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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
−7.906586555814756173109529190053, −7.45822945142754266431844267600, −6.90229339966232610613274046672, −6.13297291308769948447616071661, −5.26100921358101257249734043159, −4.14137818865433176387528873030, −3.78592832444786371141708410679, −2.94898464155020677121386096018, −1.34645643144581499011267084480, −0.57701032895185045379696906840,
0.57701032895185045379696906840, 1.34645643144581499011267084480, 2.94898464155020677121386096018, 3.78592832444786371141708410679, 4.14137818865433176387528873030, 5.26100921358101257249734043159, 6.13297291308769948447616071661, 6.90229339966232610613274046672, 7.45822945142754266431844267600, 7.906586555814756173109529190053
