| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 2·13-s + 14-s + 16-s − 18-s − 8·19-s + 21-s − 4·23-s + 24-s − 2·26-s − 27-s − 28-s − 6·29-s + 4·31-s − 32-s + 36-s − 10·37-s + 8·38-s − 2·39-s − 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.218·21-s − 0.834·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 1.29·38-s − 0.320·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.10098223614986, −12.57845933882579, −12.12122940472866, −11.75834239572636, −11.21764122360598, −10.66115578283675, −10.50978989801675, −10.03765939870985, −9.451616406891084, −9.004189250794884, −8.516233204812917, −8.172204713902175, −7.569257929227869, −7.027972301381833, −6.543416842692684, −6.264148218071806, −5.675605089581749, −5.280229267427041, −4.485640303371756, −3.976571092670515, −3.611701907998729, −2.789168900489424, −2.189000254090805, −1.688725098679849, −1.049075765306317, 0, 0,
1.049075765306317, 1.688725098679849, 2.189000254090805, 2.789168900489424, 3.611701907998729, 3.976571092670515, 4.485640303371756, 5.280229267427041, 5.675605089581749, 6.264148218071806, 6.543416842692684, 7.027972301381833, 7.569257929227869, 8.172204713902175, 8.516233204812917, 9.004189250794884, 9.451616406891084, 10.03765939870985, 10.50978989801675, 10.66115578283675, 11.21764122360598, 11.75834239572636, 12.12122940472866, 12.57845933882579, 13.10098223614986
