L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·11-s − 2·13-s − 14-s + 16-s + 6·17-s + 20-s + 4·22-s + 8·23-s + 25-s − 2·26-s − 28-s − 10·29-s − 8·31-s + 32-s + 6·34-s − 35-s + 2·37-s + 40-s + 2·41-s + 8·43-s + 4·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.312·41-s + 1.21·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.507957870\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.507957870\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.78455457749322615642199671558, −9.549201521661890106480717695442, −9.208458443392591964383343568093, −7.66606411347035025311400208583, −6.94538693348970336068376449402, −5.93396271986277649610423034750, −5.18898181926221620136234042615, −3.92613332099899923961417728552, −3.00315397242347340195692827616, −1.49821808996471083770615012100,
1.49821808996471083770615012100, 3.00315397242347340195692827616, 3.92613332099899923961417728552, 5.18898181926221620136234042615, 5.93396271986277649610423034750, 6.94538693348970336068376449402, 7.66606411347035025311400208583, 9.208458443392591964383343568093, 9.549201521661890106480717695442, 10.78455457749322615642199671558