| L(s) = 1 | + 0.135·2-s − 3-s − 1.98·4-s − 0.135·6-s − 1.12·7-s − 0.537·8-s + 9-s − 5.08·11-s + 1.98·12-s + 0.338·13-s − 0.151·14-s + 3.89·16-s − 1.77·17-s + 0.135·18-s − 5.13·19-s + 1.12·21-s − 0.685·22-s − 7.31·23-s + 0.537·24-s + 0.0457·26-s − 27-s + 2.22·28-s + 29-s + 8.74·31-s + 1.60·32-s + 5.08·33-s − 0.238·34-s + ⋯ |
| L(s) = 1 | + 0.0954·2-s − 0.577·3-s − 0.990·4-s − 0.0551·6-s − 0.424·7-s − 0.190·8-s + 0.333·9-s − 1.53·11-s + 0.572·12-s + 0.0939·13-s − 0.0405·14-s + 0.972·16-s − 0.429·17-s + 0.0318·18-s − 1.17·19-s + 0.245·21-s − 0.146·22-s − 1.52·23-s + 0.109·24-s + 0.00897·26-s − 0.192·27-s + 0.420·28-s + 0.185·29-s + 1.57·31-s + 0.282·32-s + 0.884·33-s − 0.0409·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5551699118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5551699118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.135T + 2T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 5.08T + 11T^{2} \) |
| 13 | \( 1 - 0.338T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 7.31T + 23T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 - 1.48T + 61T^{2} \) |
| 67 | \( 1 + 9.41T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 9.76T + 97T^{2} \) |
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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
−9.147530676922451589239639638888, −8.050416247444358702929420850645, −7.894379646578534421335950158219, −6.33373242326313554291353086860, −6.05790764408037266222091762969, −4.85179999782983522438846966953, −4.50919853439583452482543824739, −3.36305665104416248127961615640, −2.22533892996369667379036589032, −0.47509649146166903336571070911,
0.47509649146166903336571070911, 2.22533892996369667379036589032, 3.36305665104416248127961615640, 4.50919853439583452482543824739, 4.85179999782983522438846966953, 6.05790764408037266222091762969, 6.33373242326313554291353086860, 7.894379646578534421335950158219, 8.050416247444358702929420850645, 9.147530676922451589239639638888
