| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s − 2·13-s − 14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s + 20-s + 21-s − 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.926685976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.926685976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.65964829614378, −12.16141729673419, −11.81808031096815, −11.39904624909378, −10.70472742079708, −10.27010225178630, −9.855967275250828, −9.575237707953625, −9.163609044612283, −8.406788020804552, −8.056136890689738, −7.884645933896156, −7.170617365909875, −6.727594430125987, −6.308275340747086, −5.540804011229950, −5.111851907059559, −4.779271214580069, −3.797052292955712, −3.462664416297062, −2.724941684505206, −2.434574102792867, −1.673493876594094, −1.110787526268300, −0.6368249477401187,
0.6368249477401187, 1.110787526268300, 1.673493876594094, 2.434574102792867, 2.724941684505206, 3.462664416297062, 3.797052292955712, 4.779271214580069, 5.111851907059559, 5.540804011229950, 6.308275340747086, 6.727594430125987, 7.170617365909875, 7.884645933896156, 8.056136890689738, 8.406788020804552, 9.163609044612283, 9.575237707953625, 9.855967275250828, 10.27010225178630, 10.70472742079708, 11.39904624909378, 11.81808031096815, 12.16141729673419, 12.65964829614378
