L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 5.05·11-s − 12-s − 0.622·13-s + 14-s + 15-s + 16-s − 7.18·17-s + 18-s + 2.42·19-s − 20-s − 21-s + 5.05·22-s + 23-s − 24-s + 25-s − 0.622·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.52·11-s − 0.288·12-s − 0.172·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.74·17-s + 0.235·18-s + 0.557·19-s − 0.223·20-s − 0.218·21-s + 1.07·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s − 0.122·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.655424667\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.655424667\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 29 | \( 1 - 9.80T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 + 7.67T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 0.622T + 97T^{2} \) |
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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
−8.155109973224860013378891800475, −7.35354470110653714438906915856, −6.48555533124232623760208199573, −6.37619576812260405567399370058, −5.16092490036845980527988391723, −4.51155620739594390068174826934, −4.04989640627433137809718045818, −3.02564302027663880066865190222, −1.92280795109423715630708697016, −0.859338200676367680233421972024,
0.859338200676367680233421972024, 1.92280795109423715630708697016, 3.02564302027663880066865190222, 4.04989640627433137809718045818, 4.51155620739594390068174826934, 5.16092490036845980527988391723, 6.37619576812260405567399370058, 6.48555533124232623760208199573, 7.35354470110653714438906915856, 8.155109973224860013378891800475