L(s) = 1 | − 11-s − 2·13-s + 6·17-s + 4·23-s − 2·29-s + 10·37-s − 6·41-s + 8·43-s − 4·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s − 4·67-s − 12·71-s + 14·73-s + 16·79-s − 12·83-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.834·23-s − 0.371·29-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 1.63·73-s + 1.80·79-s − 1.31·83-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.115367578\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.115367578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.66825713621318, −15.04162545771013, −14.55435216552812, −14.23441915701990, −13.40356850989767, −12.87990022156605, −12.51218826900467, −11.76436278368414, −11.33632724014619, −10.68195563838813, −10.01871225229276, −9.623178397889004, −9.039826488889793, −8.187480776521842, −7.787257256361616, −7.214651836525235, −6.511082407945337, −5.802972746926757, −5.211713616676850, −4.662099592292870, −3.794959262065611, −3.106252640028647, −2.476586398044074, −1.492737076914549, −0.6243313477528390,
0.6243313477528390, 1.492737076914549, 2.476586398044074, 3.106252640028647, 3.794959262065611, 4.662099592292870, 5.211713616676850, 5.802972746926757, 6.511082407945337, 7.214651836525235, 7.787257256361616, 8.187480776521842, 9.039826488889793, 9.623178397889004, 10.01871225229276, 10.68195563838813, 11.33632724014619, 11.76436278368414, 12.51218826900467, 12.87990022156605, 13.40356850989767, 14.23441915701990, 14.55435216552812, 15.04162545771013, 15.66825713621318