| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 3·11-s + 12-s − 4·13-s − 14-s + 16-s + 17-s − 18-s − 5·19-s + 21-s + 3·22-s + 8·23-s − 24-s + 4·26-s + 27-s + 28-s − 4·29-s − 3·31-s − 32-s − 3·33-s − 34-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 − 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.218·21-s + 0.639·22-s + 1.66·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 0.742·29-s − 0.538·31-s − 0.176·32-s − 0.522·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.675299926820458907960167041905, −7.65614210673823424287945617515, −7.39663595426932394401833020915, −6.43280594667941399976275638484, −5.28997812483306955981466340282, −4.64442128938351027349581850001, −3.34911579752608238714976404720, −2.52173983419986001949599491683, −1.63403627755713416807467330243, 0,
1.63403627755713416807467330243, 2.52173983419986001949599491683, 3.34911579752608238714976404720, 4.64442128938351027349581850001, 5.28997812483306955981466340282, 6.43280594667941399976275638484, 7.39663595426932394401833020915, 7.65614210673823424287945617515, 8.675299926820458907960167041905
