| L(s) = 1 | + 2-s − 3-s + 4-s + 2.85·5-s − 6-s + 2.30·7-s + 8-s + 9-s + 2.85·10-s − 3.88·11-s − 12-s − 5.31·13-s + 2.30·14-s − 2.85·15-s + 16-s − 4.68·17-s + 18-s − 19-s + 2.85·20-s − 2.30·21-s − 3.88·22-s − 2.42·23-s − 24-s + 3.12·25-s − 5.31·26-s − 27-s + 2.30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.27·5-s − 0.408·6-s + 0.869·7-s + 0.353·8-s + 0.333·9-s + 0.901·10-s − 1.17·11-s − 0.288·12-s − 1.47·13-s + 0.614·14-s − 0.736·15-s + 0.250·16-s − 1.13·17-s + 0.235·18-s − 0.229·19-s + 0.637·20-s − 0.502·21-s − 0.827·22-s − 0.505·23-s − 0.204·24-s + 0.625·25-s − 1.04·26-s − 0.192·27-s + 0.434·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 + 5.31T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 + 0.677T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 0.702T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 - 9.90T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 0.151T + 83T^{2} \) |
| 89 | \( 1 + 7.05T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.60407130210873599718018441645, −6.78175252562182472529981723929, −6.19935925461832473765488718165, −5.37008953848570137187918417745, −4.94197471847514560000582793562, −4.49711509040238199440260387461, −3.09358967077827886081647335824, −2.14722082416973317471123828010, −1.79363429059189686039444752035, 0,
1.79363429059189686039444752035, 2.14722082416973317471123828010, 3.09358967077827886081647335824, 4.49711509040238199440260387461, 4.94197471847514560000582793562, 5.37008953848570137187918417745, 6.19935925461832473765488718165, 6.78175252562182472529981723929, 7.60407130210873599718018441645
