L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 4·11-s + 16-s + 6·17-s − 20-s − 4·22-s + 23-s + 25-s + 8·29-s + 8·31-s + 32-s + 6·34-s − 2·37-s − 40-s + 2·41-s + 8·43-s − 4·44-s + 46-s + 50-s + 4·55-s + 8·58-s + 10·59-s + 8·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.852·22-s + 0.208·23-s + 1/5·25-s + 1.48·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.158·40-s + 0.312·41-s + 1.21·43-s − 0.603·44-s + 0.147·46-s + 0.141·50-s + 0.539·55-s + 1.05·58-s + 1.30·59-s + 1.01·62-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
\(L(1)\) |
\(\approx\) |
\(4.157160208\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.157160208\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.82067101117147, −13.16178582731896, −12.84424901427535, −12.19468549164148, −11.95485031069477, −11.49189021176816, −10.67761713602586, −10.42551514564438, −10.03288409515324, −9.346207527092248, −8.574170721979360, −8.177963905855147, −7.602153446216760, −7.360321942382347, −6.531021064370572, −6.096652518395604, −5.453466902753250, −5.008713084695343, −4.509310376864363, −3.896919720213071, −3.153585230132874, −2.827906007740027, −2.214590314538277, −1.177424233763756, −0.6230359755676898,
0.6230359755676898, 1.177424233763756, 2.214590314538277, 2.827906007740027, 3.153585230132874, 3.896919720213071, 4.509310376864363, 5.008713084695343, 5.453466902753250, 6.096652518395604, 6.531021064370572, 7.360321942382347, 7.602153446216760, 8.177963905855147, 8.574170721979360, 9.346207527092248, 10.03288409515324, 10.42551514564438, 10.67761713602586, 11.49189021176816, 11.95485031069477, 12.19468549164148, 12.84424901427535, 13.16178582731896, 13.82067101117147