L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 6·13-s + 16-s + 2·17-s − 20-s + 23-s + 25-s + 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s + 10·37-s + 40-s − 6·41-s − 8·43-s − 46-s + 8·47-s − 50-s − 6·52-s + 6·53-s + 6·58-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.158·40-s − 0.937·41-s − 1.21·43-s − 0.147·46-s + 1.16·47-s − 0.141·50-s − 0.832·52-s + 0.824·53-s + 0.787·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4861279516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4861279516\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.77457959509221, −13.07115688203561, −12.68264934199069, −12.24438001191652, −11.66343825777461, −11.32181899921255, −10.82305267431184, −10.08749292371411, −9.853969747140141, −9.335552241526568, −8.804788471260228, −8.233584776054083, −7.692009184517690, −7.233066249361155, −7.017516665652421, −6.214416289312326, −5.437549487209360, −5.236225568023602, −4.371120881507728, −3.839126961028055, −3.137673384077453, −2.524713803425435, −1.959376108488721, −1.169676411140891, −0.2577818088736915,
0.2577818088736915, 1.169676411140891, 1.959376108488721, 2.524713803425435, 3.137673384077453, 3.839126961028055, 4.371120881507728, 5.236225568023602, 5.437549487209360, 6.214416289312326, 7.017516665652421, 7.233066249361155, 7.692009184517690, 8.233584776054083, 8.804788471260228, 9.335552241526568, 9.853969747140141, 10.08749292371411, 10.82305267431184, 11.32181899921255, 11.66343825777461, 12.24438001191652, 12.68264934199069, 13.07115688203561, 13.77457959509221