L(s) = 1 | + 9.52·3-s + 5·5-s − 21.2·7-s + 63.7·9-s + 56.1·11-s − 14.1·13-s + 47.6·15-s − 3.13·17-s + 34.2·19-s − 202.·21-s + 23·23-s + 25·25-s + 349.·27-s + 62.4·29-s − 163.·31-s + 535.·33-s − 106.·35-s + 79.7·37-s − 134.·39-s + 327.·41-s + 7.29·43-s + 318.·45-s + 238.·47-s + 107.·49-s − 29.9·51-s + 299.·53-s + 280.·55-s + ⋯ |
L(s) = 1 | + 1.83·3-s + 0.447·5-s − 1.14·7-s + 2.36·9-s + 1.54·11-s − 0.300·13-s + 0.819·15-s − 0.0447·17-s + 0.413·19-s − 2.10·21-s + 0.208·23-s + 0.200·25-s + 2.49·27-s + 0.399·29-s − 0.949·31-s + 2.82·33-s − 0.512·35-s + 0.354·37-s − 0.551·39-s + 1.24·41-s + 0.0258·43-s + 1.05·45-s + 0.738·47-s + 0.313·49-s − 0.0821·51-s + 0.775·53-s + 0.688·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.209045753\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.209045753\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
good | 3 | \( 1 - 9.52T + 27T^{2} \) |
| 7 | \( 1 + 21.2T + 343T^{2} \) |
| 11 | \( 1 - 56.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.13T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 29 | \( 1 - 62.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 163.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 79.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.29T + 7.95e4T^{2} \) |
| 47 | \( 1 - 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 299.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 3.32T + 2.05e5T^{2} \) |
| 61 | \( 1 + 492.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 991.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 808.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 567.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−9.109809208952291750667850289715, −8.293739877082100110136676602516, −7.26844214980501888358426195314, −6.79863670943955831876048596302, −5.85243608905956428662649909853, −4.39681802164160675634319611721, −3.66950036009418864871012301274, −2.97195685766851163459761858838, −2.09658902797136495759215113018, −1.03101445916997453789252368899,
1.03101445916997453789252368899, 2.09658902797136495759215113018, 2.97195685766851163459761858838, 3.66950036009418864871012301274, 4.39681802164160675634319611721, 5.85243608905956428662649909853, 6.79863670943955831876048596302, 7.26844214980501888358426195314, 8.293739877082100110136676602516, 9.109809208952291750667850289715