| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 2·9-s − 10-s − 11-s − 12-s + 15-s + 16-s − 2·17-s − 2·18-s + 3·19-s − 20-s − 22-s + 23-s − 24-s − 4·25-s + 5·27-s − 29-s + 30-s − 3·31-s + 32-s + 33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.213·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.962·27-s − 0.185·29-s + 0.182·30-s − 0.538·31-s + 0.176·32-s + 0.174·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380926 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380926 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.87992632389550, −12.36240011074374, −11.98443041558469, −11.63925906419804, −11.26348348956986, −10.83018240524912, −10.39922602226408, −9.900666490130549, −9.286149848886631, −8.843969905357051, −8.207046407540293, −7.937138685358278, −7.313704308628127, −6.823227311545792, −6.461013757221365, −5.860907353815297, −5.436555045710808, −4.984100844757660, −4.699126497472349, −3.813936740301004, −3.577882779481547, −2.979370077488155, −2.428455590709394, −1.748471781643464, −1.152147168602019, 0, 0,
1.152147168602019, 1.748471781643464, 2.428455590709394, 2.979370077488155, 3.577882779481547, 3.813936740301004, 4.699126497472349, 4.984100844757660, 5.436555045710808, 5.860907353815297, 6.461013757221365, 6.823227311545792, 7.313704308628127, 7.937138685358278, 8.207046407540293, 8.843969905357051, 9.286149848886631, 9.900666490130549, 10.39922602226408, 10.83018240524912, 11.26348348956986, 11.63925906419804, 11.98443041558469, 12.36240011074374, 12.87992632389550
