| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 2·11-s + 12-s + 13-s − 2·14-s + 16-s − 17-s − 18-s − 8·19-s + 2·21-s + 2·22-s − 24-s − 26-s + 27-s + 2·28-s − 6·29-s + 6·31-s − 32-s − 2·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.83·19-s + 0.436·21-s + 0.426·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.31849588300008, −14.78807358864999, −14.41406510357575, −13.66290117625983, −13.22675155144856, −12.63821133420728, −12.10574036428782, −11.41623612171966, −10.82311486733435, −10.54237500455287, −9.984824002850299, −9.094374175210044, −8.880859878697407, −8.262930999260067, −7.785642356949014, −7.343383282225082, −6.548207623267127, −6.041565566847808, −5.281826411526067, −4.499682551917193, −4.025772697478621, −3.148163495492574, −2.310154038353045, −1.991464389366869, −1.050603660221127, 0,
1.050603660221127, 1.991464389366869, 2.310154038353045, 3.148163495492574, 4.025772697478621, 4.499682551917193, 5.281826411526067, 6.041565566847808, 6.548207623267127, 7.343383282225082, 7.785642356949014, 8.262930999260067, 8.880859878697407, 9.094374175210044, 9.984824002850299, 10.54237500455287, 10.82311486733435, 11.41623612171966, 12.10574036428782, 12.63821133420728, 13.22675155144856, 13.66290117625983, 14.41406510357575, 14.78807358864999, 15.31849588300008
