L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·11-s + 14-s + 16-s + 6·17-s + 4·22-s − 23-s − 28-s + 8·29-s − 8·31-s − 32-s − 6·34-s + 2·37-s − 2·41-s − 8·43-s − 4·44-s + 46-s + 49-s + 56-s − 8·58-s − 10·59-s + 8·62-s + 64-s + 12·67-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.852·22-s − 0.208·23-s − 0.188·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 0.603·44-s + 0.147·46-s + 1/7·49-s + 0.133·56-s − 1.05·58-s − 1.30·59-s + 1.01·62-s + 1/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.38813445015829, −13.86610565060580, −13.32320716955819, −12.74052426756904, −12.31542471539410, −11.92803449199394, −11.19033043194696, −10.71725285951048, −10.27479551329028, −9.787886600524276, −9.466293441225762, −8.662726987204276, −8.174654277098522, −7.839380387848225, −7.222252126302715, −6.732097993788414, −6.049703665206864, −5.497031758940894, −5.065805937331373, −4.283401022698161, −3.347077294793955, −3.104146456029607, −2.326300752424492, −1.615175892375658, −0.7956285151624908, 0,
0.7956285151624908, 1.615175892375658, 2.326300752424492, 3.104146456029607, 3.347077294793955, 4.283401022698161, 5.065805937331373, 5.497031758940894, 6.049703665206864, 6.732097993788414, 7.222252126302715, 7.839380387848225, 8.174654277098522, 8.662726987204276, 9.466293441225762, 9.787886600524276, 10.27479551329028, 10.71725285951048, 11.19033043194696, 11.92803449199394, 12.31542471539410, 12.74052426756904, 13.32320716955819, 13.86610565060580, 14.38813445015829