| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 3·11-s − 12-s − 2·13-s + 16-s + 3·17-s − 18-s − 2·19-s + 3·22-s − 23-s + 24-s − 5·25-s + 2·26-s − 27-s + 3·29-s − 2·31-s − 32-s + 3·33-s − 3·34-s + 36-s + 2·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.639·22-s − 0.208·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s + 0.557·29-s − 0.359·31-s − 0.176·32-s + 0.522·33-s − 0.514·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6757482584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6757482584\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.85685753896999708559830378436, −7.47181243798406890745391233988, −6.61841481950924087145336006233, −5.91802471433505871368439399175, −5.27877849239575325214079249181, −4.50840592287740248532031725419, −3.48905775033346669475190770010, −2.54823519120828756753341658571, −1.69503852156441659711408210279, −0.47537976874356447709779393385,
0.47537976874356447709779393385, 1.69503852156441659711408210279, 2.54823519120828756753341658571, 3.48905775033346669475190770010, 4.50840592287740248532031725419, 5.27877849239575325214079249181, 5.91802471433505871368439399175, 6.61841481950924087145336006233, 7.47181243798406890745391233988, 7.85685753896999708559830378436
