| L(s) = 1 | + 2-s − 1.13·3-s + 4-s + 5-s − 1.13·6-s − 1.31·7-s + 8-s − 1.70·9-s + 10-s − 0.369·11-s − 1.13·12-s − 5.82·13-s − 1.31·14-s − 1.13·15-s + 16-s + 6.05·17-s − 1.70·18-s + 0.866·19-s + 20-s + 1.49·21-s − 0.369·22-s − 1.64·23-s − 1.13·24-s + 25-s − 5.82·26-s + 5.35·27-s − 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.655·3-s + 0.5·4-s + 0.447·5-s − 0.463·6-s − 0.497·7-s + 0.353·8-s − 0.569·9-s + 0.316·10-s − 0.111·11-s − 0.327·12-s − 1.61·13-s − 0.351·14-s − 0.293·15-s + 0.250·16-s + 1.46·17-s − 0.402·18-s + 0.198·19-s + 0.223·20-s + 0.326·21-s − 0.0788·22-s − 0.343·23-s − 0.231·24-s + 0.200·25-s − 1.14·26-s + 1.02·27-s − 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.369T + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 19 | \( 1 - 0.866T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 0.302T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 0.473T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 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
−7.51087993806648948430459262201, −6.87868910448749272698564286014, −6.06710181249486061634453998933, −5.58466059065779839481459550956, −5.00160066510393926517600295637, −4.23586685082954097964969207389, −2.97503542510291138184186482898, −2.72703231516226233184829136066, −1.35579958479508762978357782730, 0,
1.35579958479508762978357782730, 2.72703231516226233184829136066, 2.97503542510291138184186482898, 4.23586685082954097964969207389, 5.00160066510393926517600295637, 5.58466059065779839481459550956, 6.06710181249486061634453998933, 6.87868910448749272698564286014, 7.51087993806648948430459262201
