L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5442777346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5442777346\) |
\(L(1)\) |
\(\approx\) |
\(0.5966686680\) |
\(L(1)\) |
\(\approx\) |
\(0.5966686680\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.41237257634018777525463258235, −28.74074069748193582785606567756, −28.20848177137599200332867265355, −26.78695179585501117520404698578, −25.846089064558513672904240953946, −25.01726057206836243706052778465, −23.720220737070347844788051157773, −22.62911088966144446671792527398, −21.357447911113330402595948172774, −20.581138215702908324795478962114, −18.7807827192402116476863945095, −18.360039451422438626222358284444, −17.157780274954055141462994342931, −16.37407286332616936259211297146, −15.48263461075803276701839043973, −13.44543331099180747849602988818, −12.420396615783522070073860994258, −10.989791807333251388727121383561, −10.12413741084536290344987450302, −9.28098016147305560033383054835, −7.50605990952547477844201961950, −6.26951687272234859133900119159, −5.50395450485475334881882691478, −2.999351933413840279224430062353, −1.16426932189043392075392276036,
1.16426932189043392075392276036, 2.999351933413840279224430062353, 5.50395450485475334881882691478, 6.26951687272234859133900119159, 7.50605990952547477844201961950, 9.28098016147305560033383054835, 10.12413741084536290344987450302, 10.989791807333251388727121383561, 12.420396615783522070073860994258, 13.44543331099180747849602988818, 15.48263461075803276701839043973, 16.37407286332616936259211297146, 17.157780274954055141462994342931, 18.360039451422438626222358284444, 18.7807827192402116476863945095, 20.581138215702908324795478962114, 21.357447911113330402595948172774, 22.62911088966144446671792527398, 23.720220737070347844788051157773, 25.01726057206836243706052778465, 25.846089064558513672904240953946, 26.78695179585501117520404698578, 28.20848177137599200332867265355, 28.74074069748193582785606567756, 29.41237257634018777525463258235